NEUTROSOPHIC OPERATIONAL RESEARCH Volume II. Editors: Prof. Florentin Smarandache Dr. Mohamed Abdel-Basset Dr. Victor Chang

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2 NEUROSOPHIC OPERIONL RESERCH Volume II Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag

3 Dedcato Dedcated wth love to our parets for the developmets of our cogtve mds, ethcal stadards, ad shared do-good values & to our beloved famles for the cotuous ecouragemet, love, ad support. ckowledgmet he book would ot have bee possble wthout the support of may people: frst, the edtors would lke to express ther apprecato to the advsory board; secod, we are very grateful to the cotrbutors; ad thrd, the revewers for ther tremedous tme, effort, ad servce to crtcally revew the varous chapters. he help of top leaders of publc ad prvate orgazatos, who spred, ecouraged, ad supported the developmet of ths book. Peer Revewers Prof. Madad Kha Departmet of Mathematcs, COMSS Isttute of Iformato echology bbottabad, Paksta. Prof. Youg Bae Ju Departmet of Mathematcs Educato, Gyeogsag Natoal Uversty Ju 588, S. Korea. Prof. Seok-Zu Sog Departmet of Mathematcs, Jeu Natoal Uversty Jeu 6343, S. Korea. Dr. Xaohog Zhag Departmet of Mathematcs, Shaax Uversty of Scece & echology, X a, 71001, P. R. Cha; or Departmet of Mathematcs, Shagha Martme Uversty, Shagha, 01306, P. R. Cha. Dr. Xlag Lag Departmet of Mathematcs, Shaax Uversty of Scece & echology, X a, 71001, P. R. Cha.

4 Neutrosophc Operatoal Research Volume II Foreword by Joh R. Edwards Preface by the edtors Pos Brussels, 017

5 Edtors: Prof. Floret Smaradache, PhD, Postdoc Mathematcs & Scece Departmet Uversty of New Mexco, US Dr. Mohamed bdel-basset Operatos Research Dept. Faculty of Computers ad Iformatcs Zagazg Uversty, Egypt Dr. Vctor Chag X a Jaotog-Lverpool Uversty Suzhou, Cha Pos Publshg House / Pos asbl Qua du Batelage, Bruxelles Belgum DP: George Lukacs ISBN

6 Neutrosophc Operatoal Research Volume II able of Cotets Foreword... 7 Preface... 9 I pplcato of Neutrosophc Optmzato echque o Mult-obectve Relablty Optmzato Model Sahdul Islam, amay Kudu II eacher Selecto Strategy Based o Bdrectoal Proecto Measure Neutrosophc Number Evromet... 9 Surapat Pramak, Rum Roy, apa Kumar Roy III New Cocept of Matrx lgorthm for MS Udrected Iterval Valued Neutrosophc Graph Sad Broum, ssa Bakal, Mohamed alea, Floret Smaradache, Kshore Kumar P K IV Optmzato of Welded Beam Structure usg Neutrosophc Optmzato echque: Comparatve Study Mrdula Sarkar, apa Kumar Roy V Mult-Obectve Neutrosophc Optmzato echque ad Its pplcato to Rser Desg Problem Mrdula Sarkar, Ptu Das, apa Kumar Roy VI russ Desg Optmzato usg Neutrosophc Optmzato echque 113 Mrdula Sarkar, apa Kumar Roy VII Mult-obectve Neutrosophc Optmzato echque ad ts pplcato to Structural Desg Mrdula Sarkar, apa Kumar Roy 5

7 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag VIII Mult-Obectve Welded Beam Optmzato usg Neutrosophc Goal Programmg echque Mrdula Sarkar, apa Kumar Roy IX Neutrosophc Modules Necat Olgu, Mkal Bal X Neutrosophc rplet Ier Product Mehmet Şah, bdullah Kargı 6

8 Neutrosophc Operatoal Research Volume II Foreword Joh R. Edwards hs book s a excellet exposto of the use of Data Evelopmet alyss (DE) to geerate data aalytc sghts to make evdece-based decsos, to mprove productvty, ad to maage cost-rsk ad beeftopportuty publc ad prvate sectors. he desg ad the cotet of the book make t a up-to-date ad tmely referece for professoals, academcs, studets, ad employees, partcular those volved strategc ad operatoal decsomakg processes to evaluate ad prortze alteratves to boost productvty growth, to optmze the effcecy of resource utlzato, ad to maxmze the effectveess of outputs ad mpacts to stakeholders. It s cocered wth the allevato of world chages, cludg chagg demographcs, acceleratg globalzato, rsg evrometal cocers, evolvg socetal relatoshps, growg ethcal ad goverace cocer, expadg the mpact of techology; some of these chages have mpacted egatvely the ecoomc growth of prvate frms, govermets, commutes, ad the whole socety. 7

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10 Neutrosophc Operatoal Research Volume II Preface Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag hs book treats all kd of data eutrosophc evromet, wth reallfe applcatos, approachg topcs as mult-obectve programmg, bdrectoal proecto method, decso-makg, teacher selecto, terval valued eutrosophc graph, score fucto, Mmum Spag ree (MS), sgle-obectve welded beam optmzato, mult-obectve Rser desg optmzato, o-lear membershp fucto, structural optmzato, geeralzed eutrosophc goal programmg, arthmetc aggregato, geometrc aggregato, welded beam desg optmzato, eutrosophc group, eutrosophc rg, eutrosophc R-module, weak eutrosophc R-module, strog eutrosophc R-module, eutrosophc R-module homomorphsm, eutrosophc trplet er product, eutrosophc trplet metrc spaces, eutrosophc trplet vector spaces, eutrosophc trplet ormed spaces, ad so o. he frst chapter (pplcato of Neutrosophc Optmzato echque o Mult-obectve Relablty Optmzato Model) proposes a mult-obectve olear relablty optmzato model takg system relablty ad system cost as two obectve fuctos. s a geeralzed verso of fuzzy set ad tutostc fuzzy set, eutrosophc set s a very useful tool to express ucertaty, mprecseess more geeral way. hus, here the authors Sahdul Islam ad amay Kudu have cosdered eutrosophc optmzato techque wth lear ad o-lear membershp fucto to solve a mult-obectve relablty optmzato model. hs proposed method s a exteso of fuzzy ad tutostc fuzzy optmzato techque whch the degree of acceptace, determacy ad reecto of obectves are smultaeously cosdered. o demostrate the methodology ad applcablty of the proposed approach, umercal examples are preseted ad evaluated by comparg the result obtaed by eutrosophc approach wth the tutostc fuzzy optmzato techque. eacher selecto strategy s a multple crtera decso-makg process volvg determacy ad vagueess, whch ca be represeted by eutrosophc umbers of the form a+ bi, where a represets determate compoet ad bi represets determate compoet. he purpose of the 9

11 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag secod chapter (eacher Selecto Strategy Based o Bdrectoal Proecto Measure Neutrosophc Number Evromet) s to develop a multple crtera group decso-makg model for teacher selecto strategy based o bdrectoal proecto measure based method eutrosophc umber evromet. Seve crtera obtaed from expert opos are cosdered for selecto process. he crtera are, amely: demostrato, pedagogcal kowledge, acto research, emotoal stablty, kowledge o chld phychology, socal qualty, ad leadershp qualty. Weghts of the decso makers are cosdered as equal. he bdrectoal proecto measure of eutrosophc umbers s a useful mathematcal tool that ca deal wth decso-makg problems wth determate formato. Usg the bdrectoal proecto measure, a ew mult crtera decso-makg strategy s proposed. Usg bdrectoal proecto measures betwee each alteratve ad the deal alteratve, all the alteratves are preferece-raked to select the best oe. Fally, teacher selecto problem for secodary educato s solved, example by authors Surapat Pramak, Rum Roy ad apa Kumar Roy, demostratg the applcablty ad effectveess of the developed bdrectoal proecto strategy. I the thrd chapter ( New Cocept of Matrx lgorthm for MS Udrected Iterval Valued Neutrosophc Graph), Sad Broum, ssa Bakal, Mohamed alea, Floret Smaradache ad Kshore Kumar P K troduce a ew algorthm for fdg a mmum spag tree (MS) of a udrected eutrosophc weghted coected graph whose edge weghts are represeted by a terval valued eutrosophc umber. I addto, the authors compute the cost of MS ad compare the de-eutrosophed value wth a equvalet MS havg the deteremstc weghts. Fally, a umercal example s provded. he fourth chapter (Optmzato of Welded Beam Structure usg Neutrosophc Optmzato echque: Comparatve Study) vestgates Neutrosophc Optmzato (NSO) approach to optmze the cost of weldg of a welded steel beam, whle the maxmum shear stress the weld group, maxmum bedg stress the beam, maxmum deflecto at the tp ad bucklg load of the beam have bee cosdered as flexble costrats. he problem of desgg a optmal welded beam cossts of dmesog a welded steel beam ad the weldg legth so as to mmze ts cost, subect to the costrats as stated above. Specfcally based o truth, determacy ad falsty membershp fucto, a sgle obectve NSO algorthm has bee developed by authors Mrdula Sarkar ad apa Kumar Roy to optmze the weldg cost, subected to a set of flexble costrats. It s show that NSO s a effcet method fdg out the optmum value comparso to other teratve methods for olear welded beam desg precse ad mprecse evromet. Numercal example s also gve to demostrate the effcecy of the proposed NSO approach. 10

12 Neutrosophc Operatoal Research Volume II he ffth chapter (Mult-Obectve Neutrosophc Optmzato echque ad Its pplcato to Rser Desg Problem) ams to gve computatoal algorthm to solve a mult-obectve o-lear programmg problem (MONLPP) usg eutrosophc optmzato method. he proposed method s for solvg MONLPP wth sgle valued eutrosophc data. comparatve study of optmal soluto has bee made betwee tutostc fuzzy ad eutrosophc optmzato techque. he developed algorthm has bee llustrated by a umercal example. Fally, optmal rser desg problem s preseted by authors Mrdula Sarkar, Ptu Das ad apa Kumar Roy as a applcato of such techque. Mrdula Sarkar ad apa Kumar Roy develop the sxth chapter (russ Desg Optmzato usg Neutrosophc Optmzato echque) a eutrosophc optmzato (NSO) approach for optmzg the desg of plae truss structure wth sgle obectve subect to a specfed set of costrats. I ths optmum desg formulato, the obectve fuctos are the weght of the truss ad the deflecto of loaded ot; the desg varables are the cross-sectos of the truss members; the costrats are the stresses members. classcal truss optmzato example s preseted to demostrate the effcecy of the eutrosophc optmzato approach. he test problem cludes a two-bar plaar truss subected to a sgle load codto. hs sgle-obectve structural optmzato model s solved by fuzzy ad tutostc fuzzy optmzato approach as well as eutrosophc optmzato approach. Numercal example s gve to llustrate our NSO approach. he result shows that the NSO approach s very effcet fdg the best dscovered optmal solutos. he seveth chapter, called Mult-obectve Neutrosophc Optmzato echque ad ts pplcato to Structural Desg, s authored also by Mrdula Sarkar ad apa Kumar Roy, who develop a mult-obectve o-lear eutrosophc optmzato (NSO) approach for optmzg the desg of plae truss structure wth multple obectves subect to a specfed set of costrats. I ths optmum desg formulato, the obectve fuctos are the weght of the truss ad the deflecto of loaded ot; the desg varables are the cross-sectos of the truss members; the costrats are the stresses members. classcal truss optmzato example s preseted to demostrate the effcecy of the eutrosophc optmzato approach. he test problem cludes a three-bar plaar truss subected to a sgle load codto. hs mult-obectve structural optmzato model s solved by eutrosophc optmzato approach. Wth lear ad o-lear membershp fucto. he eght chapter (Mult-Obectve Welded Beam Optmzato usg Neutrosophc Goal Programmg echque) vestgates mult obectve 11

13 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Neutrosophc Goal Optmzato (NSGO) approach to optmze the cost of weldg ad deflecto at the tp of a welded steel beam, whle the maxmum shear stress the weld group, maxmum bedg stress the beam, ad bucklg load of the beam have bee cosdered as costrats. he problem of desgg a optmal welded beam cossts of dmesog a welded steel beam ad the weldg legth so as to mmze ts cost, subect to the costrats as stated above. he classcal welded bream desg structure s preseted to demostrate the effcecy of the eutrosophc goal programmg approach. he model s umercally llustrated by geeralzed NSGO techque wth dfferet aggregato method. he result shows that the Neutrosophc Goal Optmzato techque s very effcet fdg the best optmal solutos. he th chapter (Neutrosophc Modules) attempts to study the eutrosophc modules ad eutrosophc submodules. Neutrosophc logc s a exteso of the fuzzy logc whch determacy s cluded. Neutrosophc Sets are a sgfcat tool of descrbg the completeess, determacy, ad cosstecy of the decso-makg formato. Modules are oe of fudemetal ad rch algebrac structure wth respect to some bary operato the study of algebra. I ths chapter, the authors Necat Olgu ad Mkal Bal study some basc defto of eutrosophc R-modules, ad eutrosophc submodules algebra are geeralzed. Some propertes of eutrosophc R- modules ad eutrosophc submodules are preseted. he authors use classcal modules ad eutrosophc rgs. Cosequetly, they troduce eutrosophc R- modules, whch s completely dfferet from the classcal module the structural propertes. lso, eutrosophc quotet modules ad eutrosophc R-module homomorphsm are explaed, ad some deftos ad theorems are gve. Fally, some useful examples are gve to verfy the valdty of the proposed deftos ad results. I the teth ad last chapter (Neutrosophc rplet Ier Product), a oto of eutrosophc trplet er product s gve ad propertes of eutrosophc trplet er product spaces are studed. he eutrosophc trplets ad eutrosophc trplet structures were troduced by Smaradache ad l Furthermore, the authors Mehmet Şah ad bdullah Kargı also show that ths eutrosophc trplet oto s dfferet from the classcal oto. 1

14 Neutrosophc Operatoal Research Volume II I pplcato of Neutrosophc Optmzato echque o Mult-obectve Relablty Optmzato Model Sahdul Islam 1 amay Kudu * 1, Departmet of Mathematcs, Uversty of Kalya, Kalya, 74135, West Begal, Ida. * Correspodg author. E-mal: tamaykudu.math@gmal.com bstract I ths paper, we propose a mult-obectve o-lear relablty optmzato model takg system relablty ad system cost as two obectve fuctos. s a geeralzed verso of fuzzy set ad tutostc fuzzy set, eutrosophc set s a very useful tool to express ucertaty, mprecseess more geeral way. hus, here we have cosdered eutrosophc optmzato techque wth lear ad o-lear membershp fucto to solve ths mult-obectve relablty optmzato model. hs proposed method s a exteso of fuzzy ad tutostc fuzzy optmzato techque whch the degree of acceptace, determacy ad reecto of obectves are smultaeously cosdered. o demostrate the methodology ad applcablty of the proposed approach, umercal examples are preseted ad evaluated by comparg the result obtaed by eutrosophc approach wth the tutostc fuzzy optmzato techque at the ed of the paper. Keywords Relablty, Mult-obectve programmg, Neutrosophc set, Neutrosophc optmzato. 1 Itroducto I 1965, Zadeh [1] frst troduced the cocept of fuzzy set. he fuzzy set theory whch cosders the degree of membershp of elemets, s a very effectve tool to measure ucertaty real lfe stuato. I recet tme, the fuzzy set 13

15 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag theory has bee wdely developed ad geeralzed form have appeared. Itutostc fuzzy set (IFS) theory s oe of the geeralzed versos of fuzzy set theory. I 1986, taassov [] exteded the cocept of fuzzy set ad troduced tutostc fuzzy set theory, whch cosder ot oly the degree of membershp but also the degree of o-membershp fucto such that the sum of both values s less tha oe. s a geeralzato of fuzzy set theory [1], tutostc fuzzy set theory [11], terval valued fuzzy sets [9] etc., eutrosophc sets (NSs) was frst troduced by Smaradache 1995 [3]. I real world stuatos we ofte ecouter wth complete, determate ad cosstet formato, eutrosophc set s a powerful mathematcal tool to deal wth them. Neutrosophc sets whch s characterzed by a truth membershp fucto, a determacy membershp fucto ad a falsty membershp fucto, cotas both the real stadard ad o-stadard tervals ad thus t s very dffcult to apply NSs practcal feld such as real scetfc ad egeerg applcatos. I order to use NSs real lfe applcato, Wag et al. [4] proposed sgle valued eutrosophc sets (SVNSs) ad also troduced the cocept of terval valued eutrosophc sets (IVNS) [5], whch s more realstc, precse ad flexble tha SVNSs. Relablty egeerg s oe of the mportat tasks desgg ad developmet of a techcal system. he prmary goal of the relablty egeer has bee always to fd the best way to crease system relablty. he dversty of system resources, resource costrats ad optos for relablty mprovemet lead to the costructo ad aalyss of several optmzato models. I daly lfe, due to some ucertaty udgemets of the decso maker (DM), there are some coeffcets ad parameters the optmzato model, whch are always mprecse wth vague ature. I order to hadle such type of ature multobectve optmzato model, fuzzy approach s use to evaluate ths. Park [6] frst appled fuzzy optmzato techques to the problem of relablty apportomet for a seres system. Rav et al. [7] used fuzzy global optmzato relablty model. Huag [8] preseted a mult obectve fuzzy optmzato method to relablty optmzato problem. Later, tutostc fuzzy optmzato method s also appled to varous feld of research work. Sharma [10] proposed a method to aalyse the etwork system relablty whch s based o tutostc fuzzy set theory. Jaa ad Roy [13] descrbed tutostc fuzzy lear programmg method trasportato problems. lso, Mahapatra [14] troduced tutostc fuzzy mult obectve mathematcal programmg o relablty optmzato model. Nowadays eutrosophc optmzato techque s a ope feld of research work. Roy [15] appled eutrosophc lear programmg approach to mult 14

16 Neutrosophc Operatoal Research Volume II obectve producto plag problem. Pramak [16] dscussed the framework of eutrosophc mult obectve lear programmg problem. Baset et al. [17,- 30] troduced goal programmg eutrosophc evromet. I ths paper we have troduced a fuzzy mult-obectve relablty optmzato model whch system relablty ad cost of the system are cosdered as two obectve fuctos. hs s very frst whe eutrosophc optmzato techque s appled o mult-obectve o-lear relablty optmzato model. he motvato of the preset study s to gve a computatoal procedure for solvg mult-obectve relablty optmzato model by eutrosophc optmzato approach to fd the optmal soluto whch maxmze the system relablty ad mmze the cost of the system. lso as a applcato of the proposed optmzato techque to a relablty model of LCD, dsplay ut s preseted. he results of the proposed approach are evaluated by comparg wth tutostc fuzzy optmzato (IFO) techque at the ed of the paper. Mathematcal model Let R be the relablty of the th compoet of a system ad R S (R) represets the system relablty. Let C S (R) deote the cost of the system. Here we cosder a complex system, whch cludes a fve-stage combato relablty model. Fg 1: Relablty model of a LCD dsplay ut..1. Relablty model of a LCD dsplay ut Now we are terested to fd out the system relablty of a LCD dsplay ut [1] whch cossts of several compoet coected to oe aother. hs complex system maly cossts fve stages L, ( = 1,,...,5), whch are seres. hus, the geeralzed formula for the system relablty of the proposed model s gve by 15

17 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag where R S (R) = L 1 L L 3 L 4 L 5 = =1 L (.1.1) L 1 : LCD pael wth relablty R 1,.e. L 1 = R 1 ; L : backlghtg board wth 10 bulbs wth dvdual bulb relablty R such that the board fuctos wth at most oe bulb falure,.e. L = R R 9 (1 R ); L 3 : wo Mcroprocessor boards ad B hooked up parallel, each wth relablty R 3,.e. L 3 = 1 (1 R 3 ) ; L 4 : Dual power supples stadby redudacy, each power supply wth relablty R 4,.e. L 4 = R 4 + R 4 l (1 R 4 ) ; L 5 : EMI board wth relablty R 5 hooked seres wth commo put of the power supply,.e. L 5 = R 5 ; hus we have the followg system relablty R S (R) = R 1 (R R 9 (1 R )) ( 1 (1 R 3 ) ) 5 (R 4 + R 4 l (1 R 4 )) R 5 (.1.1).. Mult-obectve Relablty Optmzato Model Here we cosder cost of the proposed complex system as a addtoal obectve fucto. Now system relablty has to be maxmzed ad cost of the system s to be mmzed subect to system space as target goal. hus, the model becomes Max R S (R) = R 1 (R R 9 (1 R )) ( 1 (1 R 3 ) ) (R 4 + R 4 l (1 R 4 ) ) R 5 M C S (R) = s. t. V S (R) = 5 =1 5 =1 c [ta ( π ) R ] α a v R V lm 0.5 R,m R 1, 0 R S 1 ; = 1,,,5 (..1) 16

18 Neutrosophc Operatoal Research Volume II where v ad c represet the space ad cost of the -th compoet of the system respectvely. V lm s the system space lmtato ad R,m s the lower boud of the relablty of each compoet. Now for smplcty of calculato ad to covert the above problem to oe type maxmzato problem, we cosder C S (R) = C S (R). hus the model (..1) have the followg form Max R S (R) Max C S (R) (..) subect to the same costrats defed (..1). 3 Prelmares Defto 3.1. (Fuzzy Set ) fuzzy set X s a set of ordered pars = {(x, μ (x)) x X}, where X s a collecto of obects deoted geercally by x ad μ (x): X [0,1] s called the membershp fucto or grade of membershp of x. Defto 3.. (Itutostc fuzzy set) tutostc fuzzy set (IFS) X, where X s the uverse of dscourse, s defed as a obect of the followg form = {< x, μ (x), ν (x) > x X}, where μ (x): X [0,1] ad ν (x): X [0,1] defed the degree of membershp ad the degree of o-membershp of the elemet x X respectvely ad for every x X, 0 μ (x) + ν (x) 1. Now for each elemet x X, the value of π (x) = 1 μ (x) ν (x) s called the degree of ucertaty of the elemet x X to the tutostc fuzzy set. Defto 3.3. (Neutrosophc set ) [18] Let X be a space of pots wth a geerc elemet X deoted by x. eutrosophc set (NS) N X s characterzed by a truth membershp fucto μ (x), a determacy membershp fucto σ (x) ad a falsty membershp fucto ν (x) ad havg of the form N = {< x μ (x), ν (x), σ (x) > x X} where μ (x), ν (x) ad σ (x) are real stadard or o-stadard subsets of ] 0,1 + [.e. 17

19 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag μ (x): X ] 0,1 + [ ν (x): X ] 0,1 + [ ad σ (x): X ] 0,1 + [. here s o restrcto o the sum of μ (x), ν (x) ad σ (x). So, 0 sup μ (x) + sup ν (x) + sup σ (x) 3 +. From the phlosophcal pot of vew, the NS takes the value from the real stadard or o-stadard subsets of ] 0,1 + [. But real lfe applcato scetfc ad egeerg problems t s dffcult to use NS wth value from the subsets of ] 0,1 + [. Ye [19] reduced NSs of o-stadards tervals to a kd of smplfed eutrosophc sets of stadard tervals that wll preserve the operatos of NSs. Defto 3.4. (Sgle-valued eutrosophc set) [0] Let X be a space of pots wth a geerc elemet x X. sgle-valued eutrosophc set (SVNS) N X s characterzed by μ (x), ν (x) ad σ (x), ad havg the form N = {< x μ (x), ν (x), σ (x) > x X} where μ (x): X [0,1] ν (x): X [0,1] ad σ (x): X [0,1] wth 0 μ (x) + ν (x) + σ (x) 3 for all x X. 4 Mathematcal alyss 4.1. Neutrosophc Optmzato echque Here we are presetg a computatoal algorthm to solve multobectve relablty optmzato model (MOROM) by sgle valued eutrosophc optmzato (NSO) approach ad the followg steps are used Computatoal algorthm Step 1: mult-obectve o-lear programmg takg k obectve fuctos ca be take as - Maxmze ( f 1 (x), f (x), f k (x)) 18

20 Neutrosophc Operatoal Research Volume II Subect to g (x) b, = 1,,, m; x 0; (4.1.1) Step : Solve the above mult-obectve o-lear programmg model (4.1.1) takg oly oe obectve fucto at a tme ad avod the others, so that we ca get the deal solutos. Wth the values of all obectve fuctos evaluated at these deal solutos, the pay-off matrx ca be formulated as follows- able 1: Pay-off matrx of the soluto of k sgle obectve o lear programmg problem. Step 3: Determe the upper boud ad lower boud for each obectve fucto as follows U r = max {f r ( x 1 ), f r (x ),., f r (x k )} r = 1,,., k ad L r = m {f r ( x 1 ), f r (x ),., f r (x k )} r = 1,,., k (4.1.) So, L r f r (x) U r Where U r ad L r are respectvely upper ad lower bouds for truth membershp of the r-th obectve fucto f r (x), r. Step 4: Now the upper ad lower bouds for determacy ad falsty membershp of obectves ca be preseted as follows U r I = U r ad L r I = U r t 1 (U r L r ) U r F = U r t (U r L r ) ad L r F = L r r. (4.1.3) Where U r I, L r I are upper ad lower bouds for determacy membershp ad U r F, L r F are upper ad lower bouds for falsty membershp of the r-th obectve fucto f r (x). Here t 1 ad t are two real parameters les betwee 0 ad 1. 19

21 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Step 5: Costruct the truth membershp, determacy membershp ad falsty membershp fuctos as follows () Lear membershp fucto (B) No-lear membershp fucto 0

22 Neutrosophc Operatoal Research Volume II where Ψ ad δ r are two o-zero parameters prescrbed by the decso maker. Step 6: Now usg eutrosophc optmzato techque the gve mult-obectve o-lear programmg (MONLP) s equvalet to the followg o-lear problem as Max μ r (f r (x)) M ν r (f r (x)) Max σ r (f r (x)) Subect to μ r (f r ) ν r (f r ), μ r (f r ) σ r (f r ), ν r (f r ) 0, 0 μ r (f r ) + σ r (f r ) + ν r (f r ) 3, g (x) b, = 1,,, m; x 0, r = 1,,., k (4.1.10) where μ r (f r ), σ r (f r ) ad ν r (f r ) are the truth membershp fucto, determacy membershp fucto ad falsty membershp fucto of eutrosophc decso set respectvely. Step 7: Now usg addtve operator, the above problem (4.1.10) s reduced to the followg crsp model Maxmze k r=1 {μ r (f r ) ν r (f r ) + σ r (f r )} subect to the same costrats descrbed (4.1.10). (4.1.11) 1

23 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Step 8: Solve (4.1.1 ) to get optmal soluto. 4.. Neutrosophc Optmzato techque o Mult-obectve Relablty Optmzato problem o solve the above defed problem (..), pay-off matrx s formulated as follows - Now the best upper boud ad worst lower boud are detfed. he upper ad lower boud for truth membershp fucto of the obectve fuctos are defed as U RS = max { R S (R 1 ), R S (R )} U CS = max {C S (R 1 ), C S (R ) } L RS = m { R S (R 1 ), R S (R )} L CS = m {C S (R 1 ), C S (R ) (4..1) where L RS R S (R) U RS ad L CS C S (R) U CS lso the upper ad lower bouds for determacy ad falsty membershp of obectve fuctos ca be preseted as U R S I = U RS ad L R S I = U R S t 1 (U R S L R S ) U C S I = U CS ad L C S I = U C S t 1 (U C S L C S ) U R S F = U R S t (U R S L R S ) ad L R S F = L R S U C S F = U C S t (U C S L C S ) ad L C S F = L C S (4..) Now the lear ad o-lear membershp fuctos are formulated for the obectve fuctos R S (R) ad C S (R). fter electg the membershp fuctos, the crsp o-lear programmg problem s formulated as follows

24 Neutrosophc Operatoal Research Volume II Maxmze [μ RS (R S (R)) + μ CS (C S (R)) ν RS (R S (R)) ν CS (C S (R)) + σ RS (R S (R)) + σ CS (C S (R))] Subect to μ RS (R S ) ν RS (R S ) 5 a v R =1 V lm, μ RS (R S ) σ RS (R S ) μ CS (C S ) ν CS (C S ) μ CS (C S ) σ CS (C S ) 0 μ RS (R S ) + ν RS (R S ) + σ RS (R S ) 3 0 μ CS (C S ) + ν CS (C S ) + σ CS (C S ) 3 ν RS (R S ) 0, ν CS (C S ) 0 (4..3) 0.5 R,m R 1, 0 R S 1 ; = 1,,,5 Solve the above crsp model to obta optmal soluto of the system relablty ad cost of the system. 5 Numercal example Now a fve-stage combato relablty model of a complex system s cosdered for umercal exposure. he problem becomes as follows: Max R S (R) = R 1 (R R 9 (1 R )) ( 1 (1 R 3 ) ) (R 4 + R 4 l (1 R 4 ) ) R 5 M C S (R) = s. t. V S (R) = 5 =1 5 =1 c [ta ( π ) R ] α a v R V lm 0.5 R,m R 1, 0 R S 1 ; = 1,,,5 (5.1) able : he put data for the MOROM (5.1) s gve as follows: c 1 c c 3 c 4 c 5 v 1 v v 3 v 4 v 5 α ( ) a ( ) V lm

25 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag able 3: Wth the above solutos pay-off matrx of the obectve fuctos s formulated as follows : Now the upper ad lower boud for truth membershp of obectve fuctos are gve by U RS = , U CS = ; L RS = , L CS = ; ad ca be wrtte as R S (R) ad C S (R) ; the upper ad lower bouds for determacy ad falsty membershp of obectve fuctos ca be preseted as U R S I = ad L R S I = t 1 ( ) U C S I = ad L C S I = t 1 ( ) U R S F = t ( ) ad L R S F = U C S F = t ( ) ad L C S F = Here we cosder t 1 = 0.00, t = ; δ 1 = 1.15, δ = 0.005; ad Ψ = 4. able 4: Comparso of optmal solutos by IFO ad NSO techque 4

26 Neutrosophc Operatoal Research Volume II he above table shows the comparso of results of the proposed approach wth the tutostc optmzato approach. It s clear from the table (4) that NSO techque wth lear membershp fucto gves more of less same system relablty ad the system cost wth the tutostc fuzzy optmzato (IFO) approach. However, perspectve of system relablty eutrosophc optmzato techque wth o-lear membershp fucto gves better result tha the IFO approach. 7 Coclusos ad Future Work Here we have troduced eutrosophc optmzato techque wth lear ad o-lear membershp fucto to fd the optmal soluto of the proposed mult-obectve o-lear relablty optmzato model. he ma am of ths paper s to gve a computatoal procedure for solvg mult-obectve relablty optmzato model by eutrosophc optmzato approach to fd the optmal soluto, whch maxmze the system relablty ad mmze the cost of the system. I table (4), the result obtaed the eutrosophc optmzato techque was compared wth the IFO method ad t shows that NSO techque wth o-lear membershp fucto gves better relable system. hus, the proposed method s a effcet ad modfed optmzato techque ad gves a hghly relable system tha the other exstg method. ckowledgemets he authors wsh to ackowledge the Uversty of Kalya for facal support through the uversty research scholarshp (URS) ad DS-PURSE the Departmet of Mathematcs, Uversty of Kalya, Kalya. Refereces [1] L..Zadeh, Fuzzy sets, Iformato ad Cotrol,8(1965), pp [] K. taassov, Itutostc fuzzy sets, Fuzzy sets ad system, 0(1986), pp [3] F. Smaradache, Neutrosophy, Neutrosophc probablty, set ad logc, mer. Res. Press, Rehoboth, US, 105p, [4] H. Wag, F. Smaradache, Y. Zhag, ad R. Suderrama. Sgle Valued Neutrosophc Sets, Multspace ad Multstructure, 4 (010) [5] H. Wag, F. Smaradache, Zhag, Y.-Q. ad R.Suderrama, Iterval Neutrosophc Sets ad Logc: heory ad pplcatos Computg, Phoex: Hexs, 005 [6] K.S.Park, Fuzzy apportomet of system relablty, IEEE rasacto o Relablty, Vol R-36, 1987,PP [7] V. Rav, P. Reddy ad H. Zmmerma, Fuzzy global optmzato of complex system relablty, IEEE rasactos o Fuzzy Systems, 8(000), pp

27 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [8] H.Z. Huag, Fuzzy mult-obectve optmzato decso makg of relablty of seres system, Mcroelectroc ad Relablty, 37(1997), pp [9] I. urkse. Iterval valued fuzzy sets based o ormal forms, Fuzzy Sets ad Systems, vol. 0 (1986) [10] M. Sharma, V. Sharma ad R. Dagwal, Relablty alyss of a System Usg Itutostc Fuzzy Sets, Iteratoal Joural of Soft Computg ad Egeerg, (01),pp [11] K. taassov, Itutostc fuzzy sets, Physca- Verlag, Hedelberg, New York. [1] H.J.Zmmerma, Fuzzy Set heory ad ts applcatos, lled Publshers lmted [13] B. Jaa ad.k.roy, Mult-obectve tutostc fuzzy lear programmg ad ts applcato trasportato model Notes o Itutostc fuzzy sets, 13(013), pp [14] G.S. Mahapatra,.K.Roy, Itutostc Fuzzy Mult-Obectve Mathematcal Programmg o Relablty Optmzato Model, Iteratoal Joural of Fuzzy Systems, 1(010), [15] R. Roy ad P.Das, Mult-Obectve Producto Plag Problem Based o Neutrosophc Lear Programmg pproach, Iteratoal Joural of Fyzzy Mathematcal rchve, (8), 015, pp [16] Pramak et al., Neutrosophc Mult-Obectve Lear Programmg, Global oural of egeerg scece ad research maagemet, 3(8): 016, pp [17] M.. Baest, M. Hezam, F. Smaradache, Neutrosophc goal programmg, Neutrosophc Sets ad Systems, 11(006), pp [18] U.Rvecco, Neutrosophc logcs: Prospects problems, Fuzzy sets ad systems, 159(008), [19] J. Ye, Mult-crtera decso makg method usg the correlato coeffcet uder sgle-value eutrosophc evromet, Iteratoal Joural of Geeral system, 4(4), 013, pp [0] J. Ye, Mult-crtera decso makg method usg aggregato operators for smplfed eutrosophc sets, Joural of Itellget ad fuzzy systems, 013, Do:10.333/IFS [1] K. Neubeck, Practcal Relablty alyss, Pearso Pretce Hall Publcato, New Jersey, 004, 1- [] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [3] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [4] bdel-basset, Mohamed, Ma Mohamed, ad ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [5] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [6] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [7] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [8] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). 6

28 Neutrosophc Operatoal Research Volume II [9] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [30] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016): [31] Mrdula Sarkar, Samr Dey, apa Kumar Roy: russ Desg Optmzato usg Neutrosophc Optmzato echque, Neutrosophc Sets ad Systems, vol. 13, 016, pp do.org/10.581/zeodo [3] Ptu Das, apa Kumar Roy: Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem, Neutrosophc Sets ad Systems, vol. 9, 015, pp do.org/10.581/zeodo [33] Moa Gamal Gafar, Ibrahm El-Heawy: Itegrated Framework of Optmzato echque ad Iformato heory Measures for Modelg Neutrosophc Varables, Neutrosophc Sets ad Systems, vol. 15, 017, pp do.org/10.581/zeodo [34] Naga Rau I, Raeswara Reddy P, Dr. Dwakar Reddy V, Dr. Krshaah G: Real Lfe Decso Optmzato Model, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [35] Wezhog Jag, Ju Ye: Optmal Desg of russ Structures Usg a Neutrosophc Number Optmzato Model uder a Idetermate Evromet, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo

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30 Neutrosophc Operatoal Research Volume II II eacher Selecto Strategy Based o Bdrectoal Proecto Measure Neutrosophc Number Evromet Surapat Pramak 1* Rum Roy apa Kumar Roy 3 1 Departmet of Mathematcs, Nadalal Ghosh B.. College, Papur, P.O.-Narayapur, Dstrct North 4 Pargaas, 74316, West Begal, Ida. E-mal: sura_pat@yahoo.co. Departmet of Mathematcs, Ida Isttute of Egeerg Scece ad echology, Shbpur, Howrah, , West Begal, Ida. E-mal: roy.rumr@gmal.com 3 Departmet of Mathematcs, Ida Isttute of Egeerg Scece ad echology, Shbpur, Howrah, , West Begal, Ida. E-mal: roy_t_k@gmal.com bstract eacher selecto strategy s a multple crtera decso-makg process volvg determacy ad vagueess, whch ca be represeted by eutrosophc umbers of the form a+ bi, where a represets determate compoet ad bi represets determate compoet. he purpose of ths study s to develop a multple crtera group decso makg model for teacher selecto strategy based o bdrectoal proecto measure based strategy eutrosophc umber evromet. Seve crtera obtaed from expert opos are cosdered for selecto process. he crtera are amely demostrato, pedagogcal kowledge, acto research, emotoal stablty, kowledge o chld phycology, socal qualty, ad leadershp qualty. Weghts of the decso makers are cosdered as equal. he bdrectoal proecto measure of eutrosophc umbers s a useful mathematcal tool that ca deal wth decso makg problems wth determate formato. Usg the bdrectoal proecto measure, a ew mult crtera decso makg strategy s proposed the artcle. Usg bdrectoal proecto measures betwee each alteratve ad the deal alteratve, all the alteratves are preferece raked to select the best oe. Fally, teacher selecto problem for secodary educato s solved to demostrate the applcablty ad effectveess of the developed bdrectoal proecto strategy. 9

31 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Keywords Neutrosophc umber, Bdrectoal Proecto method, Decso makg, eacher selecto. 1 Itroducto Multple crtera group decso makg (MCGDM) [1,, 3] addresses the problem of fdg the most desrable alteratve from all the feasble alteratves. I crsp evromet decso makers ca express ther ratgs terms of tradtoal umber or crsp umbers for classcal MCGDM [1,, 3]. deals wth crsp umbers. However, whe ucertaty volves MCGDM, the ew mathematcal tool s eeded to deal wth ucertaty. Fuzzy set [4] ad tutostc fuzzy set [5] are wdely used mathematcal tools to deal wth ostatstcal ucertaty. Itutostc fuzzy MCGDM [6, 7, 8, 9, 10, 11] strateges were reported the lterature, whch employed the fuzzy or tutostc fuzzy umbers drectly or through lgustc varables to represet the ratgs ad weghts assocated wth the problems. However, whe determacy volves as depedet etty, the fuzzy ad tutostc fuzzy caot deal such stuato. o deal determacy as depedet etty, F. Smaradache defed eutrosophc set [1]. Later sgle valued eutrosophc set (SVNS) [13], a subclass of eutrosophc set receves much popularty egeerg, scetfc ad medcal feld. Varous strateges [14-4, 71-78] for eutrosophc mult crtera decso makg (MCDM) ad MCDGM have bee reported the lterature. Varous eutrosophc hybrd structures such as rough eutrosophc set [43, 44], bpolar eutrosophc set (45), rough bpolar eutrosophc set [46], eutrosophc cubc set [47, 48], eutrosophc soft set [49], eutrosophc refed set [50], eutrosophc hestat fuzzy set [51] have bee proposed the lterature. he tred toward the study of eutrosophc theory ad applcatos ca be foud [5]. eacher selecto strategy s a MCDGM problem. G. B. Redfer [53] opoed that teacher selecto strategy paradoxcal aspect exsts. I crsp evromet, teacher selecto strategy ad related ssues have bee dscussed [54-57] I tutostc fuzzy evromet. Pramak ad Mukhopadhaya [58] developed tutostc fuzzy MCGDM strategy for teacher selecto based o grey relatoal aalyss. Motvated by the work of Pramak ad Mukhopadhaya [58], Modal ad Pramak [59] exteded t sgle valued eutrosophc evromet by employg score fucto ad accuracy fucto where eutrosophc umber s expressed as three depedet compoets represetg 30

32 Neutrosophc Operatoal Research Volume II membershp, determacy ad falsty membershp degrees. F. Smaradache [60, 61] defed eutrosophc umber (NN) aother form whch s easy to uderstad ad cogtvely sutable for workg the form r+si, where r reflects determate compoets ad si reflects determate compoet respectvely. If N = si.e. the determate compoet reaches the maxmum label, the worst stuato occurs. If N =r.e. the determate compoet vashes, the best stuato occurs. herefore, t seems that employmet of NNs s more promsg mathematcal tool to deal wth the determate ad complete formato practcal decso makg stuatos. J. Ye [6] proposed lear programmg strategy wth NN ad solved producto plag problem. I the same study, J. Ye [6] preseted some basc operatos of eutrosophc umbers ad eutrosophc umber fucto. Pramak ad Baeree [63] developed three ew eutrosophc goal programmg model to solve mult-obectve programmg problems wth eutrosophc coeffcets, where coeffcets are expressed as NNs the form r+ si. J. Ye [64] grouded a de-eutrosophcato strategy ad a possblty degree rakg strategy for NNs. I the same study, J. Ye [64] demostrated the applcablty of rakg strategy by solvg a umercal MCGDM problem. Kog, Wu, ad Ye [65] defed cose smlarty measure of NNs ad employed t to deal wth the msfre fault dagoss of gasole ege. I NNs evromet, Lu ad Lu [66] proposed MGDM based o NN geeralzed weghted power averagg operator. Zheg et al. [67] developed a MGDM method based o NN geeralzed hybrd weghted averagg operator. Lterature revew reflects that MCGDM NNs evromet s ts facy. herefore, t s ecessary to vestgate ew strategy to solve MCGDM problems NNs evromet. Proecto strateges were used for solvg MCGDM wth tutostc fuzzy formato [68, 69]. J. Ye [70] poted out the geeral proecto measures have shortcomg some cases ad eed to be mproved. I the same study, J. Ye [70] developed a bdrectoal proecto strategy for solvg MCGDM uder NNs evromet. I NNs evromet, teacher selecto strategy s yet to appear. o fll the research gap, MCGDM strategy based o bdrectoal measure s proposed for teacher selecto for secodary educato. hs study s the exteso work of Pramak ad Mukhopadhyaya [58] to NN evromet. Selecto crtera are obtaed from experts opo. For ths study, some crtera are commo to the prevous study [58] because experts agree to clude these crtera for secodary educato also. So operatoal deftos ca be foud [58]. he selected seve crtera are demostrato (C1), pedagogcal kowledge (C), acto research (C3), emotoal stablty (C4), kowledge o chld phycology (C5), socal qualty (C6) ad leadershp qualty (C7). llustratve example s solved to demostrate the feasblty ad applcablty of the proposed 31

33 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag strategy NNs evromet. I ths study, strategy s comprehesvely used cludg method, approach, process, techque, etc. Rest of the paper s orgazed the followg way. Secto descrbes some basc cocepts of NNs, the geeral proecto measure ad bdrectoal proecto measure betwee NNs. Secto 3 descrbes proecto ad bdrectoal proecto based strateges for solvg MCGDM problem. I secto 4, a llustratve example of teacher selecto strategy s preseted. Fally, secto 6 presets cocluso ad future scope of work. Prelmares I ths Secto, we provde some basc deftos that are useful the paper..1 Some cocepts of eutrosophc umbers Smaradache [13, 14, 15] frstly proposed the cocept of eutrosophc umbers whch cossts of a determate compoet ad a determate compoet ad s deoted by N= r + si where r ad s are real umbers ad I s the determacy such that I = I for > 0,0 I = 0, ad bi/ki= udefed for ay real umber k. For example, assume that N= + 3I s a NN. If I [0, 0.5], t s equvalet to N [, 3.5] for such N, ths meas that ts determate compoet s ad ts determate compoet s 3I wth the determacy I [0, 0.5] ad the possblty for the umber N s wth the terval [, 3.5]. I geeral, a NN may be cosdered as a chageable terval. Let N = r+ si be a eutrosophc umber. If r, s 0, the they are stated. Let N 1 = r 1 +s 1 I, ad N = r + s I be two eutrosophc umbers, the: N 1 + N = r 1 + r + (s 1 + s ) I; N 1 -N = r 1 -r + (s 1 -s ) I; N 1 N = r 1 r + (s 1 r + s r 1 + s 1 s ) I; (a) (b) (c) N 1 = 1 N r r r s s r r r s ) I for r 0 ad r -s (d) ( N r r r { ( ( ( r r r r 1 ( r r s ) I 1 r s ) I 1 r s ) I r s ) I 1 1 (e) 3

34 Neutrosophc Operatoal Research Volume II 33. Proecto measure of eutrosophc umbers [70] Defto.1 [70] Let R= r 1, r,,r ) ad S = (s 1, s,,s ) be two NN vectors, where r = [a + b I l, a + b I u ] ad s = [c + d I l, c + d I u ] for I [I l, I u ] ad =1,,,. he the modul of R ad S are defed as he, the proecto of the vector R o the vector S s defed as ), cos( ) ( Pr S R R R o S Here, cos(r, S) s called the cose the cluded agle betwee R ad S ad s defed as S R R S S R. ), cos(. he the modul of R ad S are defed as: R = u l I b a I b a 1 ad S = u l I d c I d c 1. ad the er product of R ad S s defed as R.S = u u l l I d c I b a I d c I b a 1 ) )( ( ) )( (. Here, cos(r, S) s called the cose measure ad s defed as S R R S S R. ), cos(. he, ), cos( ) ( Pr S R R R o S = S R.S (1) = u l u u l l I d c I d c I d c I b a I d c I b a 1 1 ) )( ( ) )( ( ().3 Bdrectoal Proecto Measure of NNs [70] hs secto presets a bdrectoal proecto measure betwee NNs. Defto 3.1 [70] Let R= r 1, r,,r ) ad S = (s 1, s,,s ) be two NN vectors, where r = [a + b I l, a + b I u ] ad s = [c + d I l, c + d I u ] for I [I l, I u ] ad =1,,,. he the modul of R ad S are defed as a

35 a Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag R l u = a b I a b I 1 ad S l u = c d I c d I ad the er product of R ad S s defed as R.S = l l u u ( a b I )( c d I ) ( a b I )( c d I ). he the bdrectoal proecto 1 measure betwee R ad S s defed as: 1 Bpro(R, S)= 1 1 R. S R. S S R = R S R S R S. (3) R. S Obvously, the closure the value of Bpro(R, S) s to 1, the closure R s to S. Bpro(R, S) = 1 f ad oly f R = S. 0 Bpro(R, S) 1 for ay two NN vector R ad S whch s a ormalzed measure. Defto 3. [70] Let R= ad S= be two NN matrces, x k rq y k rq where x k = [a xk +b xk I l, a xk +b xk I u ] ad y k = [a yk +b yk I l, a yk +b yk I u ] for I [I l, I u ] ad k =1,,, r ; =1,,, q. he, k R r q a k 1 1 x k b x k I l a x k b x k I u, S r q a k 1 1 y k b y k I l a y k b y k I u, r q ad R.S= a k 1 1 xk b x k l I a x k b x k I u a y k b y k l I a y k b he the bdrectoal proecto measure betwee R ad S s defed as Bpro(R,S)= R S R S R S R. S y k I u (4) 3 Proecto ad Bdrectoal Proecto Based Strateges for Solvg MCGDM Problem I ths secto, we exteds work of Ye [70] to preset two strateges for MCGDM problems usg the () proecto measure, () bdrectoal proecto measure of NNs for teacher selecto. 34

36 Neutrosophc Operatoal Research Volume II For a MCGDM problem wth NNs, assume that = { 1,,, p } be a set of applcats, C = {C 1, C,, C q } be a set of crtera/attrbutes, ad D = {D 1, D,, D m } be a set of DMs or experts. If the decso maker (DM) D k (k = 1,,, m) provdes a evaluato value of the attrbute C ( = 1,,, q) for the alteratve ( = 1,,, p) by utlzg a scale from 1 (less ft) to 10 (more ft) wth determacy I that s represeted by a NN ad a, b k k R (k = 1,,, m, b k r k = alteratve decso matrx of NNs R ( = 1,,, p): r r...r r r...r R 1 = r r... r t1 t t a k + b k I, a k, b 0 k I [ I l, I u ]. he, we ca costruct the he mportace of the DMs the selecto commttee may be dfferetal decso makg stuato. For the decso makers D k (k = 1,,,m), weght vector s cosdered as: V = (v 1, v,, v m ) wth v 0 ad v 1. m 1 he weghts of attrbutes are geerally dfferet, ad the weght of the attrbute reflects the mportace of the attrbute decso makg stuato. For the attrbutes C ( = 1,,, q) the weght vector of attrbutes s cosdered as: W = (w 1, w,,w q ) wth w 0 ad w 1. q 1 () MCGDM Strategy-1 usg proecto measure (see Fg. 1): Step 1 I decso makg stuato, to perform de-eutrosophcato [70], each alteratve decso matrx of NNs X s trasformed to a equvalet alteratve decso matrx of terval umbers. trasformed to r k = [ a k + b k I l, a k + b k NN r k = a k + b k I s I u ] wth respect to the prescrbed determacy I [ I l, I u ]. Specfcato of I [ I l, I u ] depeds o decso makers choce ad eed of the practcal stuato. l u Step O calculatg s = [ s, s ]= [ w s =1,.,p) for k k k l k, w s u k ](k = 1,,m; = 1,.,q; s, we obta the weghted alteratve decso matrx k 35

37 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag s s S = s 11 1 m1 s s 1 s... s... s m 1q q... s mq Step 3 o obta the deal alteratve matrx S * =1,.,p). * * * s s... s q * * * s s... s 1 q * * s s... s m1 m We calculate * mq * s k = s, s * k u* k, =[max s l, k,max s u, k ](k =1,,m; =1,.,q; Step 4 he proecto measure betwee each weghted alteratve decso matrx ( = 1,,, p) ad the deal alteratve matrx usg equato (4) as follows:. S * S ca be calculated by * Pr ( S o S ) =. S (5) S * * S where, m q m q l u * * * S s s S k 1 1 k k k 1 1 s k s k, Step 5 he alteratves are raked a descedg order accordg to the values of Pro S * (S ) for = 1,,, p. he greater value of Pro Y * (S ) reflects the better alteratve. Step 6 Ed. 36

38 Neutrosophc Operatoal Research Volume II Fg.1: Flowchart for MCGDM strategy based o proecto measure () MCGDM Strategy- usg bdrectoal proecto measure (See fg. ): Step 1 Step 1 s the same as MCGDM strategy 1. Step Step s the same as MCGDM strategy 1. Step 3 Step 3 s the same as MCGDM strategy 1. Step 4 he bdrectoal proecto measure betwee each weghted alteratve decso matrx S ( = 1,,, p) ad the deal alteratve matrx ca be calculated by utlzg the equato (4) as follows:. * S * Bpro S, S = S S * S S S * S * S.S * (6) where, m q m q l u * * * S s s S k 1 1 k k k 1 1 s k s k, Step 5 he alteratves are raked a descedg order accordg to the values of BPro(S, S*) for = 1,,, m. he greater value of BPro(S, S*) meas the better alteratve. Step 6 Ed. 37

39 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Fg.: Flowchart for MCGDM strategy based o bdrectoal proecto measure 4 Illustratve example I ths secto, a llustratve example about teacher selecto s gve to show the applcablty of the proposed method. Suppose that a secodary sttuto s gog to recrut the post of a mathematcs teacher. fter tal screeg, fve caddates (.e. alteratves) 1,, 3, 4, 5 rema for further evaluato. commttee of fve decso makers or experts, E 1, E, E 3, E 4, E 5 has bee formed to coduct the tervew ad select the most approprate caddate. Seve crtera obtaed from expert opos, amely, demostrato (C 1 ), pedagogcal kowledge(c ), acto research(c 3 ), emotoal stablty(c 4 ), kowledge o chld phycology (C 5 ), socal qualty (C 6 ) ad leadershp qualty(c 7 ) are cosdered for selecto crtera. hus, the fve alteratve decso matrces are expressed, respectvely, as tabular form: able 1: lteratve decso matrx provded by the decso maker E 1 X 1 = 7+I 8+I 7+I 7+I 7+I 5+4I 6+I 6+I 7+I 7+I 5+4I 6+I 7+I 8+I 6+I 7+I 6+I 5+4I 5+4I 3+5I 6+I 7+I 8+I 6+I 7+I 5+4I 5+4I 7+I 7+I 7+I 6+I 7+I 7+I 7+I 6+I 38

40 Neutrosophc Operatoal Research Volume II able : lteratve decso matrx provded by the decso maker E X = 6+I 7+I 7+I 7+I 7+I 5+4I 6+I 7+I 8+I 7+I 6+I 6+I 7+I 7+I 6+I 7+I 6+I 6+I 6+I 3+5I 7+I 7+I 7+I 6+I 7+I 7+I 7+I 6+I 6+I 7+I 6+I 7+I 7+I 7+I 6+I able 3: lteratve decso matrx provded by the decso maker E 3 X 3 = 7+I 8+I 7+I 7+I 7+I 7+I 7+I 7+I 7+I 7+I 5+4I 6+I 7+I 8+I 6+I 7+I 6+I 5+4I 7+I 3+5I 6+I 5+4I 5+4I 6+I 7+I 5+4I 7+I 5+4I 7+I 6+I 6+I 7+I 7+I 5+4I 6+I able 4: lteratve decso matrx provded by the decso maker E 4 X 4 = 7+I 8+I 7+I 7+I 7+I 5+4I 6+I 6+I 7+I 7+I 5+4I 6+I 7+I 8+I 6+I 7+I 6+I 5+4I 3+5I 3+5I 6+I 7+I 8+I 6+I 7+I 5+4I 5+4I 7+I 7+I 7+I 6+I 7+I 7+I 7+I 6+I able 5: lteratve decso matrx provded by the decso maker E 5 X 5 = 8+I 8+I 7+I 5+4I 7+I 6+I 5+4I 8+I 7+I 7+I 7+I 6+I 7+I 8+I 5+4I 7+I 8+I 7+I 7+I 7+I 5+4I 5+4I 7+I 8+I 7+I 5+4I 5+4I 7+I 7+I 7+I 6+I 5+4I 7+I 5+4I 7+I he the developed strategy s appled to the decso makg problem ad descrbed by the followg steps: 39

41 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Step 1: ssume that the specfed determacy s I[0,0.5]. he each X k ca be trasferred to the followg forms: able 6: lteratve decso matrx provded by the decso maker E 1 after deeutrofcato X 1= [7,7.5] [8,8.5] [7,7.5] [7,7.5] [7,7.5] [5,7] [6,7] [6,7] [7,7.5] [7,7.5] [5,7] [6,7] [7,7.5] [8,8.5] [6,7] [7,7.5] [6,7] [5,7] [5,7] [3,5.5] [6,7] [7,7.5] [8,8.5] [6,7] [7,7.5] [5,7] [5,7] [7,7.5] [7,7.5] [7,7.5] [6,7] [7,7.5] [7,7.5] [7,7.5] [6,7] able 7: lteratve decso matrx provded by the decso maker E after deeutrofcato X = [6,7] [7,7.5] [7,7.5] [7,7.5] [7,7.5] [5,7] [6,7] [7,7.5] [8,8.5] [7,7.5] [6,7] [6,7] [7,7.5] [7,7.5] [6,7] [7,7.5] [6,7] [6,7] [6,7] [3,5.5] [7,7.5] [7,7.5] [7,7.5] [6,7] [7,7.5] [7,7.5] [7,7.5] [6,7] [6,7] [7,7.5] [6,7] [7,7.5] [7,7.5] [7,7.5] [6,7] able 8: lteratve decso matrx provded by the decso maker E 3 after deeutrofcato X 3 = [7,7.5] [8,8.5] [7,7.5] [7,7.5] [7,7.5] [7,7.5] [7,7.5] [7,7.5] [7,7.5] [7,7.5] [5,7]] [6,7] [7,7.5] [8,8.5] [6,7] [7,7.5] [6,7] [5,7] [7,7.5] [3,5.5] [6,7] [5,7] [5,7] [6,7] [7,7.5] [5,7] [7,7.5] [5,7] [7,7.5] [6,7] [6,7] [7,7.5] [7,7.5] [5,7] [6,7] 40

42 Neutrosophc Operatoal Research Volume II able 9: lteratve decso matrx provded by the decso maker E 4 after deeutrofcato X 4 = [7,7.5] [8,8.5] [7,7.5] [7,7.5] [7,7.5] [5,7] [6,7] [6,7] [7,7.5] [7,7.5] [5,7] [6,7] [7,7.5] [8,8.5] [6,7] [7,7.5] [6,7] [5,7] [5,7] [3,5.5] [6,7] [7,7.5] [8,8.5] [6,7] [7,7.5] [5,7] [5,7] [7,7.5] [7,7.5] [7,7.5] [6,7] [7,7.5] [7,7.5] [7,7.5] [6,7] able 10: lteratve decso matrx provded by the decso maker E 5 after deeutrofcato X 5 = [8,8.5] [8,8.5] [7,7.5] [5,7] [7,7.5] [6,7] [5,7] [8,8.5] [7,7.5] [7,7.5] [7,7.5] [6,7] [7,7.5] [8,8.5] [5,7] [7,7.5] [8,8.5] [7,7.5] [7,7.5] [7,7.5] [5,7] [5,7] [7,7.5] [8,8.5] [7,7.5] [5,7] [5,7] [7,7.5] [7,7.5] [7,7.5] [6,7] [5,7] [7,7.5] [5,7] [7,7.5] Step : ssume that the weghtg vector of the attrbutes s W= (0.,0.,0.,0.1,0.1,0.1,0.1). he the fve weghted decso matrces are obtaed as follows: able 11: Weghted decso matrx provded by the decso maker E 1 Y 1= [1.4,1.5] [1.6,1.7] [1.4,1.5] [.7,.75] [.7,.75] [.5,.7] [.6,.7] [1.,1.4] [1.4,1.5] [1.4,1.5] [.5,.7] [.6,.7] [.7,.75] [.8,.85] [1.,1.4] [1.4,1.5] [1.,1.4] [.5,.7] [.5,.7] [.3,.55] [.6,.7] [1.4,1.5] [1.6,1.7] [1.,1.4] [.7,.75] [.5,.7] [.5,.7] [.7,.75] [1.4,1.5] [1.4,1.5] [1.,1.4] [.7,7.5] [.7,.75] [.7,.75] [.6,.7] 41

43 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag able 1: Weghted decso matrx provded by the decso maker E Y = [1.,1.4] [1.4,1.5] [1.4,1.5] [.7,.75] [.7,.75] [.5,.7] [.6,.7] [1.4,1.5] [1.6,1.7] [1.4,1.5] [.6,.7] [.6,.7] [.7,.75] [.7,.75] [1.,1.4] [1.4,1.5] [1.,1.4] [.6,.7] [.6,.7] [.3,.55] [.7,.75] [1.4,1.5] [1.4,1.5] [1.,1.4] [.7,.75] [.7,.75] [.7,.75] [.6,.7] [1.,1.4] [1.4,1.5] [1.,1.4] [.7,.75] [.7,.75] [.7,.75] [.6,.7] able 13: Weghted decso matrx provded by the decso maker E 3 Y 3 = [1.4,1.5] [1.6,1.7] [1.4,1.5] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [1.4,1.5] [1.4,1.5] [1.4,1.5] [.5,.7] [.6,.7] [.7,.75] [.8,.85] [1.,1.4] [1.4,1.5] [1.,1.4] [.5,.7] [.7,.75] [.3,.55] [.6,.7] [1,1.4] [1,1.4] [1.,1.4] [.7,.75] [.5,.7] [.7,.75] [.5,.7] [1.4,1.5] [1.,1.4] [1.,1.4] [.7,.75] [.7,.75] [.5,.7] [.6,.7] able 14: Weghted decso matrx provded by the decso maker E 4 Y 4 = [1.4,1.5] [1.6,1.7] [1.4,1.5] [.7,.75] [.7,.75] [.5,.7] [.6,.7] [1.,1.4] [1.4,1.5] [1.4,1.5] [.5,.7] [.6,.7] [.7,.75] [.8,.85] [1.,1.4] [1.4,1.5] [1.,1.4] [.5,.7] [.5,.7] [.3,.55] [.6,.7] [1.4,1.5] [1.6,1.7] [1.,1.4] [.7,.75] [.5,.7] [.5,.7] [.7,.75] [1.4,1.5] [1.4,1.5] [1.,1.4] [.7,.75] [.7,.75] [.7,.75] [.6,.7] 4

44 Neutrosophc Operatoal Research Volume II able 15: Weghted decso matrx provded by the decso maker E 5 Y 5 = [1.6,1.7] [1.6,1.7] [1.4,1.5] [.5,.7] [.7,.75] [.6,.7] [.5,.7] [1.6,1.7] [1.4,1.5] [1.4,1.5] [.7,.75] [.6,.7] [.7,.75] [.8,.85] [1,1.4] [1.4,1.5] [1.6,1.7] [.7,.75] [.7,.75] [.7,.75] [.5,.7] [1,1.4] [1.4,1.5] [1.6,1.7] [.7,.75] [.5,.7] [.5,.7] [.7,.75] [1.4,1.5] [1.4,1.5] [1.,1.4] [.5,.7] [.7,.75] [.5,.7] [.7,.75] Step3: he deal alteratve matrx s determed as follows: Y * [1.6,1.7] [1.6,1.7] [1.,1.4] [1.4,1.5] [1.4,1.5] [1.6,1.7] [1.6,1.7] [1.4,1.5] [1.6,1.7] [1.4,1.5] [1.4,1.5] [1.4,1.5] [1.6,1.7] [1.6,1.7] [1.,1.4] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.6,.7] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.7,.75] [.8,.85] [.7,.75] [.7,.75] [.7,.75] Step4: he bdrectoal proecto measure values betwee each weghted alteratve decso matrx Y ( =1,,, 5) ad the deal alteratve matrx Y * ca be obtaed as follows: Y * , Y , Y , Y , Y , Y Y 1.Y * = 8.749, Y.Y * = , Y 3.Y * = , Y 4.Y * = , Y 5.Y * = Pro Y * (Y 1 ) = , Pro Y * (Y ) = , Pro Y * (Y 3 ) = , Pro Y * (Y 4 ) = , Pro Y * (Y 5 ) = BPro(Y 1, Y*) =0.607, BPro(Y, Y*) =0.5683, BPro(Y 3, Y*) =0.5687, BPro(Y 4, Y*) =0.6168, BPro(Y 5, Y*) = Step5: Sce, Pro Y * (Y )> Pro Y * (Y 5 )> Pro Y * (Y 4 )> Pro Y * (Y 1 )> Pro Y * (Y 3 ), the, the fve caddates are raked as: Hece, accordg to MCGDM strategy 1 based o proecto measure, s the best choce amog all caddates. 43

45 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Sce BPro(Y 5, Y * )>BPro(Y 1,Y * )>BPro(Y 4,Y * )>BPro(Y 3,Y * )>BPro(Y,Y * ), the, the fve caddates are raked as: Hece, accordg to MCGDM strategy usg bdrectoal proecto measure, 5 s the best choce amog all caddates. For I [0,.] Note. We calculate rakg order for I [0,.] ad calculatos are gve appedx. Smlarly, for specfed I rakg order ca be determed. 7 Cocluso I real teacher selecto process, determacy plays a mportat role because the experts caot preset all the crtera ad trats of applcat s completely ad accurately due to tme pressure, lack of doma kowledge, sutable evromet of tervew ad other related ssues of selecto process. So, determacy volves ther ratg ad decso. o deal such stuatos, eutrosophc umber (a + bi, where a s the determate compoet ad bi s the determate compoet) s more cogtvely effcet to preset determate ad complete formato. I ths paper, the teacher selecto procedure s studed based o proecto ad bdrectoal proecto measure. he sgfcace of the paper s that we combe NNs educatoal settg to cope wth MCGDM problems. Selecto crtera are obtaed by employg drect tervew ad opo from doma experts. he proposed teacher selecto strategy s a effectve mathematcal tool to express cogtve formato ad takg to accout the relablty of the formato. he proposed MCGDM strategy for teacher selecto smply ad relably represets huma cogto by cosderg the teractvty of crtera ad the cogto towards determacy volves the problem. he developed MCGDM strategy for teacher selecto combes the advatages of NNs ad MCGDMS, whch s more feasble ad practcal tha other strateges. he proposed strategy s farly flexble ad easy to mplemet. Proposed MCGDM strategy for teacher selecto s more comprehesve because whe I = 0, t reduces to classcal MCGDM strategy.e. crsp MCGDM strategy. lthough ths study has demostrated the effectveess of the proposed strategy, may areas eed to be explored. Future studes should address the followg problems: () the case whe determacy (I) assumes dfferet specfcatos smultaeously for ratg. () ukow weghts of the decso makers ad ukow weghts of the crtera () Other practcal decso makg where MCGDM volves. 44

46 Neutrosophc Operatoal Research Volume II ckowledgemets he authors are grateful to Prof. Dulal Mukhopahyaya, Departmet of Educato, Neta Subhas Ope Uversty, for hs ecouragemet ad dyamc support durg the study. Prof. Dulal Mukhopahyaya spet hs precous tme ad tellect to offer hs scere ad valuable suggestos coductg the study. ppedx ssume that the specfed determacy s I[0,0.] accordg to the decso makers choce ad real requremets. he, Step 1: Each X k ca be trasferred to the followg forms: able 6: lteratve decso matrx provded by the decso maker E 1 after deeutrofcato X 1= [7,7.] [8,8.] [7,7.] [7,7.] [7,7.] [5,5.8] [6,6.4] [6,6.4] [7,7.] [7,7.] [5,5.8] [6,6.4] [7,7.] [8,8.] [6,6.4] [7,7.] [6,6.4] [5,5.8] [5,5.8] [3,4] [6,6.4] [7,7.] [8,8.] [6,6.4] [7,7.] [5,5.8] [5,5.8] [7,7.] [7,7.] [7,7.] [6,6.4] [7,7.] [7,7.] [7,7.] [6,6.4] able 7: lteratve decso matrx provded by the decso maker E after deeutrofcato X = [6,6.4] [7,7.] [7,7.] [7,7.] [7,7.] [5,5.8] [6,6.4] [7,7.] [8,8.] [7,7.] [6,6.4] [6,6.4] [7,7.] [7,7.] [6,6.4] [7,7.] [6,6.4] [6,6.4] [6,6.4] [3,4] [7,7.] [7,7.] [7,7.] [6,6.4] [7,7.] [7,7.] [7,7.] [6,6.4] [6,6.4] [7,7.] [6,6.4] [7,7.] [7,7.] [7,7.] [6,6.4] 45

47 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag able 8: lteratve decso matrx provded by the decso maker E 3 after deeutrofcato X 3 = [7,7.] [8,8.] [7,7.] [7,7.] [7,7.] [7,7.] [7,7.] [7,7.] [7,7.] [7,7.] [5,5.8] [6,6.4] [7,7.] [8,8.] [6,6.4] [7,7.] [6,6.4] [5,5.8] [7,7.] [3,4] [6,6.4] [5,5.8] [5,5.8] [6,6.4] [7,7.] [5,5.8] [7,7.] [5,5.8] [7,7.] [6,6.4] [6,6.4] [7,7.] [7,7.] [5,5.8] [6,6.4] able 9: lteratve decso matrx provded by the decso maker E 4 after deeutrofcato X 4 = [7,7.] [8,8.] [7,7.] [7,7.] [7,7.] [5,5.8] [6,6.4] [6,6.4] [7,7.] [7,7.] [5,5.8] [6,6.4] [7,7.] [8,8.] [6,6.4] [7,7.] [6,6.4] [5,5.8] [3,4] [3,4] [6,6.4] [7,7.] [8,8.] [6,6.4] [7,7.] [5,5.8] [5,5.8] [7,7.] [7,7.] [7,7.] [6,6.4] [7,7.] [7,7.] [7,7.] [6,6.4] able 10: lteratve decso matrx provded by the decso maker E 5 after deeutrofcato X 5 = [8,8.] [8,8.] [7,7.] [5,5.8] [7,7.] [6,6.4] [5,5.8] [8,8.] [7,7.] [7,7.] [7,7.] [6,6.4] [7,7.] [8,8.] [5,5.8] [7,7.] [8,8.] [7,7.] [7,7.] [7,7.] [5,5.8] [5,5.8] [7,7.] [8,8.] [7,7.] [5,5.8] [5,5.8] [7,7.] [7,7.] [7,7.] [6,6.4] [5,5.8] [7,7.] [5,5.8] [7,7.] Step: ssume that the weghtg vector of the attrbutes s W=(0.,0.,0.,0.1,0.1,0.1,0.1). he the fve weghted decso matrces are obtaed as follows: 46

48 Neutrosophc Operatoal Research Volume II able 11: Weghted decso matrx provded by the decso maker E 1 Y 1= [1.4,1.44] [1.6,1.64] [1.4,1.44] [.7,.7] [.7,.7] [.5,.58] [.6,.64] [1.,1.8] [1.4,1.44] [1.4,1.44] [.5,.58] [.6,.64] [.7,.7] [.8,.8] [1.,1.8] [1.4,1.44] [1.,1.8] [.5,.58] [.5,.58] [.3,.4] [.6,.64] [1.4,1.44] [1.6,1.64] [1.,1.8] [.7,.7] [.5,.58] [.5,.58] [.7,.7] [1.4,1.44] [1.4,1.44] [1.,1.8] [.7,.7] [.7,.7] [.7,.7] [.6,.64] able 1: Weghted decso matrx provded by the decso maker E Y = [1.,1.8] [1.4,1.44] [1.4,1.44] [.7,.7] [.7,.7] [.5,.58] [.6,.64] [1.4,1.44] [1.6,1.64] [1.4,1.44] [.6,.64] [.6,.64] [.7,.7] [.7,.7] [1.,1.8] [1.4,1.44] [1.,1.8] [.6,.64] [.6,.64] [.3,.4] [.7,.7] [1.4,1.44] [1.4,1.44] [1.,1.8] [.7,.7] [.7,.7] [.7,.7] [.6,.64] [1.,1.8] [1.4,1.44] [1.,1.8] [.7,.7] [.7,.7] [.7,.7] [.6,.64] able 13: Weghted decso matrx provded by the decso maker E 3 Y 3 = [1.4,1.44] [1.6,1.64] [1.4,1.44] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [1.4,1.44] [1.4,1.44] [1.4,1.44] [.5,.58] [.6,.64] [.7,.7] [.8,.8] [1.,1.8] [1.4,1.44] [1.,1.8] [.5,.58] [.7,.7] [.3,.4] [.6,.64] [1,1.16] [1,1.16] [1.,1.8] [.7,.7] [.5,.58] [.7,.7] [.5,.58] [1.4,1.44] [1.,1.8] [1.,1.8] [.7,.7] [.7,.7] [.5,.58] [.6,.64] able 14: Weghted decso matrx provded by the decso maker E 4 Y 4 = [1.4,1.44] [1.6,1.64] [1.4,1.44] [.7,.7] [.7,.7] [.5,.58] [.6,.64] [1.,1.8] [1.4,1.44] [1.4,1.44] [.5,.58] [.6,.64] [.7,.7] [.8,.8] [1.,1.8] [1.4,1.44] [1.,1.8] [.5,.58] [.3,.4] [.3,.4] [.6,.64] [1.4,1.44] [1.6,1.64] [1.,1.8] [.7,.7] [.5,.58] [.5,.58] [.7,.7] [1.4,1.44] [1.4,1.44] [1.,1.8] [.7,.7] [.7,.7] [.7,.7] [.6,.64] 47

49 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag able 15: Weghted decso matrx provded by the decso maker E 5 Y 5 = [1.6,1.64] [1.6,1.64] [1.4,1.44] [.5,.58] [.7,.7] [.6,.64] [.5,.58] [1.6,1.64] [1.4,1.44] [1.4,1.44] [.7,.7] [.6,.64] [.7,.7] [.8,.8] [1,1.16] [1.4,1.44] [1.6,1.64] [.7,.7] [.7,.7] [.7,.7] [.5,.58] [1,1.16] [1.4,1.44] [1.6,1.64] [.7,.7] [.5,.58] [.5,.58] [.7,.7] [1.4,1.44] [1.4,1.44] [1.,1.8] [.5,.58] [.7,.7] [.5,.58] [.7,.7] Step3: he deal alteratve matrx s determed as follows: * Y [1.6,1.64] [1.6,1.64] [1.,1.8] [1.4,1.44] [1.4,1.44] [1.6,1.64] [1.6,1.64] [1.4,1.44] [1.6,1.64] [1.4,1.44] [1.4,1.44] [1.4,1.44] [1.6,1.64] [1.6,1.64] [1.,1.8] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.6,.64] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.7,.7] [.8,.8] [.7,.7] [.7,.7] [.7,.7] Step4: he proecto ad bdrectoal proecto measure values betwee each weghted alteratve decso matrx Y (=1,,,,5) ad the deal alteratve matrx Y * ca be obtaed as follows: Y = , Y 1 = 8.647, Y = , Y 3 = , Y 4 = 8.605, Y 5 = Y 1.Y * = , Y.Y * = , Y 3.Y * = , Y 4.Y * = , Y 5.Y * = Pro * Y (Y 1 ) = , Pro * Y (Y ) = , Pro * Y (Y 3 ) = , Pro * Y (Y 4 ) = , Pro * Y (Y 5 ) = BPro(Y 1, Y*) =0.5993, BPro(Y, Y*) =0.56, BPro(Y 3, Y*) =0.5146, BPro(Y 4, Y*) =0.5794, BPro(Y 5, Y*) = Step 5: Sce, Pro Y* (Y 5 )> Pro Y* (Y )> Pro Y* (Y 3 )> Pro Y* (Y 4 )> Pro Y* (Y 5 ), the, the fve caddates are raked as: Hece, accordg to MCGDM strategy 1, 5 s the best choce amog all caddates. Sce, BPro(Y 5,Y * )>BPro(Y 1,Y * )>BPro(Y 4,Y * )>BPro(Y,Y * )>BPro(Y 3,Y * ), the, the fve caddates are raked as: Hece, accordg to MCGDM strategy, 5 s the best choce amog all caddates. 48

50 Neutrosophc Operatoal Research Volume II Refereces [1] rvey, R., D., Campo, J., E., (198). he employmet tervew: summary ad revew of recet research, Persoel Psychology, 35(), [] Hwag, L., L, M., J., (1987). Group decso makg uder multple crtera: methods ad applcatos, Sprger Verlag, Hedelberg. [3] Campo, M.,., Pursell, E., D., Brow, B., K., (1988). Structured tervewg; rasg the psychometrc propertes of the employmet tervew, Persoel Psychology, 41(1), 5 4. [4] Zadeh, L. (1965). Fuzzy sets. Iformato ad Cotrol, 5, [5] taassov, K.. (1986). Itutostc fuzzy sets, Fuzzy Sets ad Systems, 0(1), [6] L, L., Yua, X., H., Xa, Z., Q., (007). Multcrtera fuzzy decso-makg based o tutostc fuzzy sets, Joural of Computers ad Systems Sceces, 73(1), [7] Bora, F. E., Gec, S., Kurt, M., kay, D. (009). mult-crtera tutostc fuzzy group decso makg for suppler selecto wth OPSIS method. Expert Systems wth pplcatos 36(8), [8] Zhag, S. F. & Lu, S. Y. (011). GR-based tutostc fuzzy mult-crtera group decso makg method for persoel selecto, Expert Systems wth pplcatos [9] Ye, J., (013). Multple attrbute group decso-makg methods wth ukow weghts tutostc fuzzy settg ad terval-valued tutostc fuzzy settg, Iteratoal Joural of Geeral Systems, 4(5), [10] Modal, K, Pramak, S. (014). Itutostc fuzzy multcrtera group decso makg approach to qualty-brck selecto problem, Joural of ppled Quattatve Methods, 9(), [11] Dey, P. P., Pramak, S., & Gr, B., C. (015). Mult-crtera group decso makg tutostc fuzzy evromet based o grey relatoal aalyss for weaver selecto Khad sttuto, Joural of ppled ad Quattatve Methods, 10(4), [1] Smaradache, F. (1998). Neutrosophy: eutrosophc probablty, set, ad logc: aalytc sythess & sythetc aalyss. Rehoboth: merca Research Press. [13] Wag, H., Smaradache F., Zhag, Y.Q., & Suderrama, R. (010). Sgle valued eutrosophc sets, Multspace ad Multstructure, 4, [14] Sodekamp, M. (013). Models, methods ad applcatos of group multplecrtera decso aalyss complex ad ucerta systems. Dssertato, Uversty of Paderbor, Germay. [15] Kharal, (014). eutrosophc mult-crtera decso makg method, New Mathematcs ad Natural Computato, 10(), [16] Bswas, P., Pramak, S., ad Gr, B. C. (014a). Etropy based grey relatoal aalyss method for mult-attrbute decso-makg uder sgle valued eutrosophc assessmets, Neutrosophc Sets ad Systems,, [17] Bswas, P., Pramak, S., ad Gr, B. C. (014b). ew methodology for eutrosophcmult-attrbute decso makg wth ukow weght formato, Neutrosophc Sets ad Systems, 3, 4-5. [18] Bswas, P., Pramak, S., & Gr, B.C. (014c). Cose smlarty measure based mult-attrbute decso-makg wth trapezodal fuzzy eutrosophc umbers, Neutrosophc Sets ad Systems, 8, [19] Broum, S., Smaradache, F. (014). Sgle valued eutrosophc trapezod lgustc aggregato operators based mult-attrbute decso makg, Bullet of Pure &ppled Sceces- Mathematcs ad Statstcs, 33(3),

51 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [0] Ye, J., & ad Zhag, Q. (014). : Sgle valued eutrosophc smlarty measures for multple attrbute decso-makg, Neutrosophc Sets ad Systems,, do.org/10.581/zeodo [1] Lu, P., Chu, Y., L, Y., ad Che, Y. (014). Some geeralzed eutrosophc umber Hamacher aggregato operators ad ther applcato to group decso makg, Iteratoal Joural of Fuzzy Systems, 16(), [] Lu, P. ad Wag, Y. (014). Multple attrbute decso-makg method based o sgle valued eutrosophc ormalzed weghted Boferro mea, Neural Computg ad pplcatos, 5(7-8), [3] Ye, J. (014). Sgle valued eutrosophc mmum spag tree ad ts clusterg method, Joural of Itellget Systems, 3(3): [4] Ye, J. (014). Multple-attrbute decso-makg method uder a sgle-valued eutrosophc hestat fuzzy evromet, Joural of Itellget Systems, DOI: /sys [5] P. Bswas, S. Pramak, & B.C. Gr. (015). Cose smlarty measure based mult-attrbute decso-makg wth trapezodal fuzzy eutrosophc umbers, Neutrosophc Sets ad Systems 8 (015), [6] Modal, K., Pramak, S., (015). Neutrosophc decso makg model of school choce, Neutrosophc Sets ad Systems, 7, [7] Modal, K., Pramak, S. (015). Neutrosophc taget smlarty measure ad ts applcato to multple attrbute decso makg, Neutrosophc Sets ad Systems, 9, [8] Ye, J. (015). rapezodal eutrosophc set ad ts applcato to multple attrbute decso-makg, Neural Computg ad pplcatos, 6, [9] J. Ye. (015). exteded OPSIS method for multple attrbute group decso makg based o sgle valued eutrosophc lgustc umbers, Joural of Itellget ad Fuzzy Systems, 8(1) (015) [30] Bswas, P., Pramak, S., & Gr, B.C. (016). OPSIS method for multattrbute group decso makg uder sgle-valued eutrosophc evromet, Neural Computg ad pplcatos 7(3), [31] Bswas, P., Pramak, S., & Gr, B. C. (016). ggregato of tragular fuzzy eutrosophc set formato ad ts applcato to mult-attrbute decso makg, Neutrosophc Sets ad Systems 1, [3] Bswas, P., Pramak, S., & Gr, B. C. (016). Value ad ambguty dex based rakg method of sgle-valued trapezodal eutrosophc umbers ad ts applcato to mult-attrbute decso makg, Neutrosophc Sets ad Systems 1, [33] Şah, R., Lu, P. Maxmzg devato method for eutrosophc multple attrbute decso makg wth complete weght formato, Neural Computg ad pplcatos, 7 (7), [34] a, Z. P., Wag, J., Zhag, H. Y., Che, X. H., & Wag, J. Q. (016). Smplfed eutrosophc lgustc ormalzed weghted Boferro mea operator ad ts applcato to mult-crtera decso-makg problems, Flomat, 30 (1): [35] Zhag, M., Lu, P., Sh, L. (016). exteded multple attrbute group decsomakg ODIM method based o the eutrosophc umbers, Joural of Itellget ad Fuzzy Systems, 30, DOI: /IFS [36] Del, I. & Y. Subas, Y. (017). rakg method of sgle valued eutrosophc umbers ad ts applcato to mult-attrbute decso makg problems, Iteratoal Joural of Mache Learg ad Cyberetcs, 8 (4), [37] Lag, R., Wag, J. & Zhag, H. (017). mult-crtera decso-makg method based o sgle-valued trapezodal eutrosophc preferece relatos wth complete weght formato, Neural Computg ad pplcatos. s

52 Neutrosophc Operatoal Research Volume II [38] Pramak, S., Bswas, P., & Gr, B. C. (017). Hybrd vector smlarty measures ad ther applcatos to mult-attrbute decso makg uder eutrosophc evromet, Neural Computg ad pplcatos, 8 (5), DOI: /s [39] Bswas, P., Pramak, S., ad Gr, B. C. (017). No-lear programmg approach for sgle-valued eutrosophc OPSIS method, New Mathematcs ad Natural Computato. (I press). [40] bdel-basset, M., Mohamed, M., & Sagaah,. K. (017). Neutrosophc HP- Delph group decso-makg model based o trapezodal eutrosophc umbers, Joural of mbet Itellgece ad Humazed Computg, /s [41] bdel-basset, M., Mohamed, M., Husse,. N., & Sagaah,. K. (017). ovel group decso-makg model based o tragular eutrosophc umbers, Soft Computg, [4] Xu, D. S., We, C., We, G. W. (017). ODIM method for sgle-valued eutrosophc multple attrbute decso-makg, Iformato, 8, 15. DOI: / fo [43] Broum, S., Smaradache, F., & Dhar, M. (014). Rough eutrosophc sets, Itala Joural of Pure ad ppled Mathematcs, 3, [44] Broum, S., Smaradache, F., & Dhar, M. (014). Rough eutrosophc sets, Neutrosophc Sets ad Systems, 3, [45] Del, I., l, M., & Smaradache, F. (015). Bpolar eutrosophc sets ad ther applcatos based o multcrtera decso makg problems, dvaced Mechatroc Systems, (ICMechs), Iteratoal Coferece, DOI: /ICMechS [46] Pramak, S. & Modal, K. (016). Rough bpolar eutrosophc set, Global Joural of Egeerg Scece ad Research Maagemet 3(6), [47] l, M., Del, I., & Smaradache, F. (016). he theory of eutrosophc cubc sets ad ther applcatos patter recogto, Joural of Itellget ad Fuzzy Systems, 30 (4) (016), [48] Ju, Y. B., Smaradache, F., & Km, C. S. (017). Neutrosophc cubc sets, New Mathematcs ad Natural Computato, 13 (1) (017), [49] Ma, P. K. (013). Neutrosophc soft set, als of Fuzzy Mathematcs ad Iformatcs, 5 (1), [50] Del, I., Broum, S. & Smaradache, F. (015b). O eutrosophc refed sets ad ther applcatos medcal dagoss, Joural of New heory, 6, [51] Ye, J. (014 b). Multple-attrbute decso-makg method uder a sglevalued eutrosophc hestat fuzzy evromet, Joural of Itellget Systems, DOI: /sys [5] Smaradache, F. & Pramak, S. (Eds). (016). New treds eutrosophc theory ad applcatos. Brussels: Pos Edtos. [53] Redfer, G. B. (1966). eacher selecto, Educatoal Leadershp, 3(7), [54] Schutteberg, E. M. (1975). Gradg teacher educato ad recrutmet, he eacher Educator, 11(3), -8. DOI: / [55] Caledo, R.J. (1986). Selectg capable teachers, ERS Spectrum 1,-6. [56] Ncholso, E. W., Mcerey, W. D. (1988). Hrg the rght teacher: a method for selecto, NSSP Bullet, 7(511), DOI: / [57] Stroge, J. H., Hdma, J. L. (006). he teacher qualty dex: a protocol for teacher selecto. ssocato for Supervso ad Currculum Developmet, lexadra, V: DOI: /

53 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [58] S. Pramak, D. Mukhopadhyaya. (011). Grey relatoal aalyss based tutostc fuzzy mult crtera group decso-makg approach for teacher selecto hgher educato, Iteratoal Joural of Computer pplcatos 34(10,), 1-9. [59] Modal, K., & Pramak, S., (014). Mult-crtera group decso makg approach for teacher recrutmet hgher educato uder smplfed eutrosophc evromet, Neutrosophc Sets ad Systems, 6, [60] Smaradache, F., (013). Itroducto to eutrosophc measure, eutrosophc tegral, ad eutrosophc Probablty, Stech & Educato Publsher, Craova Columbus. [61] Smaradache, F., (014). Itroducto to eutrosophc statstcs, Stech & Educato Publshg. [6] Ye, J. (017). Neutrosophc umber lear programmg method ad ts applcato uder eutrosophc umber evromets, Soft Computg. do: /s z. [63] Pramak, S., Baeree, D. (017). Mult-obectve lear goal programmg problem wth eutrosophc coeffcets. I F. Smaradache, & S. Pramak (Eds), New treds eutrosophc theory ad applcato. Brussels, Belgum: Pos Edtos. [64] Ye, J. (016). Multple-attrbute group decso-makg method uder a eutrosophc umber evromet, Joural of Itellget Systems, 5(3), [65] Kog, L. W., Wu, Y. F., &Ye, J. (015). Msfre fault dagoss method of gasole eges usg the cose smlarty measure of eutrosophc umbers, Neutrosophc Sets ad Systems, 8, [66] Lu, P, & Lu, X. (016). he eutrosophc umber geeralzed weghted power averagg operator ad ts applcato multple attrbute group decso makg, Iteratoal Joural of Mache Learg Cyberetcs. DOI: /s [67] Zheg, E., eg, F., & Lu, P. (017). Multple attrbute group decso-makg method based o eutrosophc umber geeralzed hybrd weghted averagg operator, Neural Computg ad pplcatos, 8 (8), [68] Yue, Z., L., (013). tutostc fuzzy proecto-based approach for parter selecto, ppled Mathematcal Modellg, 37, [69] Zeg, S., Balezets,., Che, J., ad Luo, G. (013). proecto method for multple attrbute group decso makg wth tutostc fuzzy formato, Iformatca, 4(3), [70] Ye, J., (015). Bdrectoal proecto method for multple attrbute group decso makg wth eutrosophc umber, Neural Computg ad pplcato, do: /s [71] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [7] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [73] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [74] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [75] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [76] bdel-baset, Mohamed, Ibrahm M. Hezam, ad Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [77] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015):

54 Neutrosophc Operatoal Research Volume II [78] Evely Jazmí Heríquez tepara, Oscar Omar polaro rzube, Jorge rturo Chcala rroyave, Eduardo too lvarado Uamuo, Makel Leyva Vazquez: Competeces evaluato based o sgle valued eutrosophc umbers ad decso aalyss schema, Neutrosophc Sets ad Systems, vol. 17, 017, pp [79] Slva Llaa eada Yepez: Decso support based o sgle valued eutrosophc umber for formato system proect selecto, Neutrosophc Sets ad Systems, vol. 17, 017, pp [80] W. B. Vasatha Kadasamy, K. Ilatheral, ad Floret Smaradache: Modfed Collatz coecture or (3a + 1) + (3b + 1)I Coecture for Neutrosophc Numbers, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [81] Wezhog Jag, Ju Ye: Optmal Desg of russ Structures Usg a Neutrosophc Number Optmzato Model uder a Idetermate Evromet, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [8] Jqa Che, Ju Ye: Proecto Model of Neutrosophc Numbers for Multple ttrbute Decso Makg of Clay-Brck Selecto, Neutrosophc Sets ad Systems, Vol. 1, 016, pp do.org/10.581/zeodo

55 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag III New Cocept of Matrx lgorthm for MS Udrected Iterval Valued Neutrosophc Graph Sad Broum 1,* ssa Bakal Mohamed alea 1 Floret Smaradache 3 Kshore Kumar P K 4 1 Laboratory of Iformato Processg, Faculty of Scece Be M Sk, Uversty Hassa II, B.P 7955, Sd Othma, Casablaca, Morocco. Emal: broumsad78@gmal.com, taleamohamed@yahoo.fr Ecole Royale Navale, Boulevard Sour Jdd, B.P Casablaca, Morocco. Emal: assabakal@yahoo.fr 3 Departmet of Mathematcs, Uversty of New Mexco, 705 Gurley veue, Gallup, NM 87301, US. Emal: smarad@um.edu 4 Departmet of Mathematcs, l Musaa College of echology, Sultaate of Oma. Emal: kshorepk@act.edu.om bstract I ths chapter, we troduce a ew algorthm for fdg a mmum spag tree (MS) of a udrected eutrosophc weghted coected graph whose edge weghts are represeted by a terval valued eutrosophc umber. I addto, we compute the cost of MS ad compare the de-eutrosophed value wth a equvalet MS havg the deteremstc weghts. Fally, a umercal example s provded. Keywords Iterval valued Neutrosophc Graph, Score fucto, Mmum Spag ree (MS). 1 Itroducto I order to express the cosstecy ad determacy that exst reallfe problems reasoably, Smaradache [3] proposed the cocept of eutrosophc sets (NSs) from a phlosophcal stadpot, whch s characterzed by three totally depedet fuctos,.e., a truth-fucto, a determacy fucto ad a falsty fucto that are sde the real stadard or o-stadard ut terval ]-0, 1+[. Hece, eutrosophc sets ca be regarded as may exteded forms of classcal fuzzy sets [8] such as tutostc fuzzy sets [6], terval-valued 54

56 Neutrosophc Operatoal Research Volume II tutostc fuzzy sets [7] etc. Moreover, for the sake of applyg eutrosophc sets real-world problems effcetly, Smaradache [9] put forward the oto of sgle valued eutrosophc sets (SVNSs for short) frstly, ad the varous theoretcal operators of sgle valued eutrosophc sets were defed by Wag et al. [4]. Based o sgle valued eutrosophc sets, Wag et al. [5] further developed the oto of terval valued eutrosophc sets (IVNSs for short), some of ther propertes were also explored. Sce the, studes of eutrosophc sets ad ther hybrd extesos have bee pad great atteto by umerous scholars [19]. May researchers have proposed a frutful results o terval valued eutrosophc sets [1,14,16,17,18,0,1-31] MS s most fudametal ad well-kow optmzato problem used etworks graph theory. he obectve of ths MS s to fd the mmum weghted spag tree of a weghted coected graph. It has may real tme applcatos, whch cludes commucato system, trasportato problems, mage processg, logstcs, wreless etworks, cluster aalyss ad so o. he classcal problems related to MS [1], the arc legths are take ad t s fxed so that the decso maker use the crsp data to represet the arc legths of a coected weghted graph. But the real world scearos the arch legth represets a parameter whch may ot have a precse value. For example, the demad ad supply, cost problems, tme costrats, traffc frequeces, capactes etc., For the road etworks, eve though the geometrc dstace s fxed, arc legth represets the vehcle travel tme whch fluctuates due to dfferet weather codtos, traffc flow ad some other uexpected factors. here are several algorthms for fdg the MS classcal graph theory. hese are based o most well-kow algorthms such as Prms ad Kruskals algorthms. Nevertheless, these algorthms caot hadle the cases whe the arc legth s fuzzy whch are take to cosderato []. More recetly, some scholars have used eutrosophc methods to fd mmum spag tree eutrosophc evromet. Ye [8] defed a method to fd mmum spag tree of a graph where odes (samples) are represeted the form of NSs ad dstace betwee two odes represets the dssmlarty betwee the correspodg samples. Madal ad Basu [9] defed a ew approach of optmum spag tree problems cosderg the cosstecy, completeess ad determacy of the formato. hey cosdered a etwork problem wth multple crtera represeted by weght of each edge eutrosophc sets. Kadasamy [11] proposed a double-valued Neutrosophc Mmum Spag ree (DVN-MS) clusterg algorthm to cluster the data represeted by double-valued eutrosophc formato. Mulla [15] dscussed the MS problem o a graph whch a bpolar eutrosophc umber s assocated to each 55

57 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag edge as ts edge legth, ad llustrated t by a umercal example. o the ed, o research dealt wth the cases of terval valued eutrosophc arc legths. he ma obectve of ths work s to fd the mmum spag tree of udrected eutrosophc graphs usg the proposed matrx algorthm. It would be very much useful ad easy to hadle the cosdered problem of terval valued eutrosophc arc legths usg ths algorthm. he rest of the paper s orgazed as follows. Secto brefly troduces the cocepts of eutrosophc sets, sgle valued eutrosophc sets, terval valued eutrosophc sets ad the score fucto of terval valued eutrosophc umber. Secto 3 proposes a ovel approach for fdg the mmum spag tree of terval valued eutrosophc udrected graph. I Secto 4, two llustratve examples are preseted to llustrate the proposed method. Fally, Secto 5 cotas coclusos ad future work. Prelmares Defto.1 [3] Le uversal set be a uversal set. he eutrosophc set o the categorzed to three membershp fuctos called the true ( x ), determate I( x) ad false F ( x) cotaed real stadard or ostadard subset of ] - 0, 1 + [ respectvely. 0 sup ( x) + sup I ( x ) + sup I ( x) 3 + (1) Defto. [4] Let be a uversal set. he sgle valued eutrosophc sets (SVNs) o the uversal s deoted as followg = {<x: ( x ), I( x ), F ( x ) > x } () he fuctos ( x) [0. 1], I( x) [0. 1] ad F ( x) [0. 1] are amed degree of truth, determacy ad falsty membershp of x, satsfy the followg codto: 0 ( x) + I ( x ) + ( x) 3 (3) Defto.3 [5]. terval valued eutrosophc set X s defed as a obect of the form ~ ~ ~ x, ~ t,, f : x X, where ~ t L U ~, ~ ~, I L I U ~, ~, f F L F U ~, ~ ~, 56

58 Neutrosophc Operatoal Research Volume II ad L ~ degree where, U ~ L ~, L I ~ U ~, U I ~ L I ~,, L F~ U I ~,, U F~ L F~, :X U F~ [0, 1].he terval membershp deotes the lower ad upper truth membershp, lower ad upper determate membershp ad lower ad upper false membershp of a elemet X correspodg to a terval valued p p p eutrosophc set where 0 I F 3 I order to rak the IVNS, N [18] defed the followg score fucto. ~ Defto.4 [18]. Let ~ ~ ~ t,, f be a terval valued t L U ~, ~ I L I U ~, ~ ~ f F L F U ~, ~,he, eutrosophc umber, where ~ ~,, the score fucto s ( ), accuracy fucto a IVNN ca be represeted as follows: a ( ) M M M ad certaty fucto c ( ) of () S N L L L U U U ~ ( ~ I ~ F~ ) ( ~ I ~ F~ ) ( ), 6 ~ S ( ) 0,1 (4) () a N L L U U ~ ( ~ F~ ) ( ~ F~ ) ~ ( ) a ( ) 1,1 (5) N [18] gave a order relato betwee two IVNNs, whch s defed as follows ~ Let ~ ~ ~ ~ ~ ~ 1 t1, 1, f1 ad ~ t,, f be two terval valued eutrosophc umbers the. If s( 1) s( ), the deoted by 1 1 s greater tha, that s, 1s superor to. If s( 1) s( ),ad a( 1) a( ) the 1 s greater tha, that s, 1 s superor to, deoted by 1. If s( 1) s( ), a( 1) a( ), the 1 s equal to, that s, 1 s dfferet to, deoted by 1 Defto.5 [17]: Let L, U, I L, I U, F L, F U be a IVNN, the score fucto S of s defed as follows 1 L U L U L L SRIDVN I I F F, S 1,1. (6) 4, 57

59 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag L U L U L U Defto.6 [17]: Let,, I, I, F, F be a IVNN, the accuracy fucto H of s defed as follows L U 1 H RIDVN, H 1,1. U U L L U L L U I (1 ) I (1 ) F (1 I ) F (1 I ) (7) L U L U L U o rak ay two IVNNs,, I, I, F, F ad L U L U L U B,, I, I, F, F, scores of B B B B B B Rdva [17] troduced the followg method. Defto.7 [17]: Let ad B be two IVNNs, ad B H respectvely, ad S ad S B be ad HB be accuracy values of ad B respectvely, the. If S( ) S( B) the s larger tha B, deoted B.. If S( ) S( B) the we check ther accuracy values ad decde as follows: (a) If H( ) H( B), the B. (b) However, f H( ) H( B), the s larger tha B, deoted B. L U L U L U Defto.8 [1]: Let,, I, I, F, F be a IVNN, the score fucto S of s defed as follows L U L U L L L U L L I I F F F F 4 4 SNNCY, S 0,1. 8 Remark.9: I eutrosophc mathematcs, the zero sets are represeted by the followg form 0 N ={<x, [0, 0], [1, 1], [1,1])> :x X}. 3 he proposed algorthm he followg algorthm s a ew cocept of fdg the MS of udrected terval valued eutrosophc graph usg the matrx approach. lgorthm: Iput: the weght matrx M = W for the udrected weghted terval valued eutrosophc graph G. Output: Mmum cost Spag tree of G. 58

60 Neutrosophc Operatoal Research Volume II Step 1: Iput terval valued eutrosophc adacecy matrx. Step : Covert the terval valued eutrosophc matrx to a score matrx S usg the score fucto. Step 3: Iterate step 4 ad step 5 utl all (-1) etres matrx of S are ether marked or set to zero or other words all the ozero elemets are marked. Step 4: Fd the weght matrx M ether colums-wse or row-wse to determe the umarked mmum etres correspodg edge e M. Step 5: If the correspodg edge e of selected prevous marked etres of the score matrx S the set S whch s the weght of the S produces a cycle wth the S = 0 else mark S Step 6: Costruct the graph cludg oly the marked etres from the score matrx S whch shall be desred mmum cost spag tree of G. 4 Practcal example 4.1 Example 1 I ths secto, a umercal example of IVNMS s used to demostrate of the proposed algorthm. Cosder the followg graph G= (V, E) show Fgure 1, wth fves odes ad seve edges. Varous steps volved the costructo of the mmum cost spag tree are descrbed as follow. Fg.1. Udrected terval valued eutrosophc graphs 59

61 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag able 1. e e 1 Edge legth <[.3,.4], [.1,.], <[.,.4]> e 13 e 14 e 4 e 34 e 35 e 45 <[.4,.5], [.,.6], <[.4,.6]> <[.1,.3], [.6,.8], <[.8,.9]> <[.4,.5], [.8,.9], <[.3,.4]> <[.,.4], [.3,.4], <[.7,.8]> <[.4,.5], [.6,.7], <[.5,.6]> <[.5,.6], [.4,.5], <[.3,.4] he terval valued eutrosophc adacecy matrx s computed below: = 0 e 1 e 13 e 14 0 e e 4 0 e e 34 e 35 e 14 e 4 e 34 0 e 45 [ 0 0 e 35 e 45 0 ] pplyg the score fucto proposed by a [18], we get the score matrx: S= [ ] I ths matrx, the mmum etres 0.17 s selected ad the correspodg edge (1, 4) s marked by the gree color. Repeat the procedure utl termato (Fgure ). 60

62 Neutrosophc Operatoal Research Volume II Fg.. Marked terval valued eutrosophc graphs he ext o-zero mmum etres 0.4 s marked ad correspodg edges (3, 5) are also colored (Fgure 3) S = [ ] Fg. 3. Marked terval valued eutrosophc graphs ext terato S = [ ] he ext o-zero mmum etres 0.45 s marked. he correspodg marked edges are portrayed Fgure 4. 61

63 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Fg. 4. Marked terval valued eutrosophc graphs ext terato S= [ ] he ext o-zero mmum etres 0. 5 s marked. he correspodg marked edges are portrayed Fgure 5. Fg. 5. Marked terval valued eutrosophc graphs ext terato S= [ ] 6

64 Neutrosophc Operatoal Research Volume II he ext mmum o-zero elemet s marked. However, whle drawg the edges, t produces the cycle so we delete ad mark t as 0 stead of (Fgure 6). Fg. 6. Cycle {1, 3, 4} S= [ ] he ext mmum o-zero elemet s marked. However, whle drawg the edges, t produces the cycle so we delete ad mark t as 0 stead of (Fgure 7). Fg. 7. Cycle {3, 4, 5} S= [ ] 63

65 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag he ext mmum o-zero elemet s marked. However, whle drawg the edges, t produces the cycle so we delete ad mark t as 0 stead of (Fgure 8). Fg. 8. Marked edges the ext roud Fally, we get the fal path of mmum cost of spag tree of G s portrayed Fgure 9. Fg. 9. Fal path of mmum cost of spag tree of the graph d thus, the crsp mmum cost spag tree s ad the fal path of mmum cost of spag tree s{, 4},{4, 1},{4, 3},{3, 5}. he procedure s termato. 4. Example he score fucto s used mache learg volved mapulatg probabltes. Here the score fuctos the proposed algorthm plays a vtal role detfyg the mmum spag tree of udrected terval valued eutrosophc graphs. lso based o the order of polyomal tme computato the score fucto used are approachg towards dfferet MS for a Neutrosophc graph. We compare our proposed method wth these scorg methods used by dfferet researchers ad hece compute the MS of udrected terval valued eutrosophc graphs. 64

66 65 Neutrosophc Operatoal Research Volume II

67 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag 5 Comparatve study I what follows we compare the proposed method preseted secto 4 wth other exstg methods cludg the algorthm proposed by Mulla et al [15] as follow Iterato 1: Let C 1 = {1} ad C1 ={, 3, 4,5} Iterato : Let C ={1, 4} ad C ={, 3,5} Iterato 3: Let C 3 ={1, 4, 3} ad C 3 ={, 5} Iterato 4: Let C 4 ={1,3, 4, 5} ad C4 ={} Fally, the terval valued eutrosophc mmal spag tree s Fg.10. IVN mmal spag tree obtaed by Mulla s algorthm. So, t ca be see that the terval valued eutrosophc mmal spag tree {, 4},{4, 1},{4, 3},{3, 5}.obtaed by Mulla s algorthm, fter deeutrosophcato of edges weght usg the score fucto, s the same as the path obtaed by proposed algorthm. he dfferece betwee the proposed algorthm ad Mulla s algorthm s that our approach s based o Matrx approch, whch ca be easly mplemeted Matlab, whereas the Mulla s algorthm s based o the comparso of edges each terato of the algorthm ad ths leads to hgh computato. 66

68 Neutrosophc Operatoal Research Volume II 7 Coclusos ad Future Work hs artcle aalyse about the mmum spag tree problem where the edges weghts are represeted by terval valued eutrosophc umbers. I the proposed algorthm, may examples were vestgated o MS. he ma obectve of ths study s to focus o algorthmc approach of MS ucerta evromet by usg eutrosophc umbers as arc legths. I addto, the algorthm we use s smple eough ad more effectve for real tme evromet. hs work could be exteded to the case of drected eutrosophc graphs ad other kds of eutrosophc graphs such as bpolar ad terval valued bpolar eutrosophc graphs. I future, the proposed algorthm could be mplemeted to the real tme scearos trasportato ad supply cha maagemet the feld of operatos research. O the other had, graph terpretatos (decso trees) of syllogstc logcs ad bezer curves eutrosophc world could be cosdered ad mplemeted as the real-lfe applcatos of atural logcs ad geometres of data [31-36]. ckowledgemets he authors are very grateful to the edtor of chapter book ad revewers for ther commets ad suggestos, whch s helpful mprovg the paper. Refereces [1] Madal, Dutta J, Pal S.C. ew effcet techque to costruct a mmum spag tree. Iteratoal Joural of dvaced Research Computer Scece ad software Egeerg, Issue 10, 01, pp [] Dey, Pal, Prm's algorthm for solvg mmum spag tree problem fuzzy evromet. als of Fuzzy Mathematcs ad Iformatcs. [3] Smaradache F. Neutrosophy. Neutrosophc Probablty, Set, ad Logc, ProQuest Iformato & Learg. rbor, Mchga, US, 1998,105 p. [4] Wag H, Smaradache F, Zhag Y, ad Suderrama R. Sgle valued eutrosophc sets. Multsspace ad Multstructure. 010; 4: [5] Wag H, Smaradache F, Zhag Y. Q, ad Suderrama R. Iterval eutrosophc sets ad logc: heory ad pplcatos Computg, Hexs, Phoex, Z, 005. [6] taassov K, Gargov G. Iterval valued tutostc fuzzy sets. Fuzzy Sets ad Systems. 1989; vol.31: [7] Zadeh L. Fuzzy sets. Iform ad Cotrol. 1965, 8: [8] Kadasamy I. Double-Valued Neutrosophc Sets, ther Mmum Spag rees, ad Clusterg lgorthm. Joural of Itellget system 016, pp [9] Ye J. sgle valued eutrosophc mmum spag tree ad ts clusterg method. Joural of Itellget system 3, 014, pp [10] Madal K ad Basu K. Improved smlarty measure eutrosophc evromet ad ts applcato fdg mmum spag tree. Joural of Itellget & Fuzzy System 31, 016, pp

69 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [11] Nacy & H. Garg, mproved score fucto for rakg eutrosophc sets ad ts applcato to decso makg process, Iteratoal Joural for Ucertaty Quatfcato, 6 (5), 016, pp [1] Broum S, alea M, Bakal, Smaradache F. Iterval Valued Neutrosophc Graphs. Crtcal Revew, XII, 016. pp.5-33 [13] Broum S, Smaradache F, alea M., Bakal. Decso-makg method based o the terval valued eutrosophc graph. Future echologe, 016, IEEE, pp [14] Mulla M, Broum S, Stephe. Shortest path problem by mmal spag tree algorthm usg bpolar eutrosophc umbers. Iteratoal Joural of Mathematc reds ad echology, 017, Vol 46, N. pp [16] Ye J. Iterval Neutrosophc Multple ttrbute Decso-Makg Method wth Credblty Iformato. teratoal Joural of Fuzzy Systems. 016:1-10. [17] Şah R ad Lu PD. Maxmzg devato method for eutrosophc multple attrbute decso makg wth complete weght formato, Neural Computg ad pplcatos. (015), DOI: /s [18] ag G. pproaches for Relatoal Multple ttrbute Decso Makg wth Iterval Neutrosophc Numbers Based o Choquet Itegral. Master hess, Shadog Uversty of Face ad Ecoomcs, Ja, Cha, [19] Del I, Şubaş Y, Smaradache F ad l M. Iterval valued bpolar eutrosophc sets ad ther applcato patter recogto. 016 IEEE Iteratoal Coferece o Fuzzy Systems (FUZZ-IEEE) 016, pp [0] Broum S, Smaradache F, alea M ad Bakal. Operatos o Iterval Valued Neutrosophc Graphs. chapter book- New reds Neutrosophc heory ad pplcatos- Floret Smaradache ad Surpat Pramak (Edtors), 016, pp ISBN [1] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [3] bdel-basset, Mohamed, Ma Mohamed, ad ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [4] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [5] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [6] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [7] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [8] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [9] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016): [30] M.. Baest, M. Hezam, F. Smaradache, Neutrosophc goal programmg, Neutrosophc Sets ad Systems, 11(006), pp [31] opal, S. ad Öer,. Cotrastgs o extual Etalmetess ad lgorthms of Syllogstc Logcs, Joural of Mathematcal Sceces, Volume, Number 4, prl 015, pp

70 Neutrosophc Operatoal Research Volume II [3] opal S. Obect-Oreted pproach to Couter-Model Costructos Fragmet of Natural Laguage, BEU Joural of Scece, 4(), , 015. [33] opal, S. Syllogstc Fragmet of Eglsh wth Dtrastve Verbs Formal Sematcs, Joural of Logc, Mathematcs ad Lgustcs ppled Sceces, Vol 1, No 1, 016. [34] opal, S. ad Smaradache, F. Lattce-heoretc Look: Negated pproach to dectval (Itersectve, Neutrosophc ad Prvate) Phrases. he 017 IEEE Iteratoal Coferece o INovatos Itellget Sysems ad pplcatos (INIS 017); (accepted for publcato). [35] aş, F. ad opal, S. Bezer Curve Modelg for Neutrosophc Data Problem. Neutrosophc Sets ad Systems, Vol 16, pp. 3-5, [36] opal, S. Equvaletal Structures for Bary ad erary Syllogstcs, Joural of Logc, Laguage ad Iformato, Sprger, 017, DOI: /s [37] S. Broum,. Dey,. Bakal, M. alea, F. Smaradache, L. H. So, D. Koley: Uform Sgle Valued Neutrosophc Graphs, Neutrosophc Sets ad Systems, vol. 17, 017, pp [38]. Chalapath, R. V M S S Kra Kumar: Neutrosophc Graphs of Fte Groups, Neutrosophc Sets ad Systems, vol. 15, 017, pp do.org/10.581/zeodo [39] Nasr Shah, Sad Broum: Irregular Neutrosophc Graphs, Neutrosophc Sets ad Systems, vol. 13, 016, pp do.org/10.581/zeodo [40] Nasr Shah: Some Studes Neutrosophc Graphs, Neutrosophc Sets ad Systems, vol. 1, 016, pp do.org/10.581/zeodo [41] Sad Broum, ssa Bakal, Mohamed alea, Floret Smaradache: Isolated Sgle Valued Neutrosophc Graphs, Neutrosophc Sets ad Systems, vol. 11, 016, pp do.org/10.581/zeodo

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72 Neutrosophc Operatoal Research Volume II IV Optmzato of Welded Beam Structure usg Neutrosophc Optmzato echque: Comparatve Study Mrdula Sarkar 1 apa Kumar Roy 1 Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah, , Ida. Emal: mrdula.sarkar86@redffmal.com Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah, , Ida. Emal: emal-roy_t_k@yahoo.co. bstract hs paper vestgates Neutrosophc Optmzato (NSO) approach to optmze the cost of weldg of a welded steel beam, whle the maxmum shear stress the weld group, maxmum bedg stress the beam, maxmum deflecto at the tp ad bucklg load of the beam have bee cosdered as flexble costrats. he problem of desgg a optmal welded beam cossts of dmesog a welded steel beam ad the weldg legth so as to mmze ts cost, subect to the costrats as stated above. he purpose of the preset study frstly to vestgate the effect of truth, determacy ad falsty membershp fucto NSO perspectve of welded beam desg mprecse evromet ad secodly s to aalyse the results obtaed by dfferet optmzato methods lke fuzzy, tutostc fuzzy ad several determstc methods so that the weldg cost of the welded steel beam become most cost effectve. Specfcally based o truth, determacy ad falsty membershp fucto, a sgle obectve NSO algorthm has bee developed to optmze the weldg cost, subected to a set of flexble costrats. It has bee show that NSO s a effcet method fdg out the optmum value comparso to other teratve methods for olear welded beam desg precse ad mprecse evromet. Numercal example s also gve to demostrate the effcecy of the proposed NSO approach. 71

73 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Keywords Neutrosophc Set, Sgle Valued Neutrosophc Set, Neutrosophc Optmzato, Sgle-Obectve welded beam optmzato. 1 Itroducto I today s hghly compettve market, the pressure o a costructo agecy s to fd better ways to atta the optmal soluto. I covetoal optmzato problems, t s assumed that the decso maker s sure about the precse values of data volved the model. However, real world applcatos all the parameters of the optmzato problems may ot be kow precsely due to ucotrollable factors. Such type of mprecse data s well represeted by fuzzy umber troduced by Zadeh [1]. I realty, a decso maker may assume that a obect belogs to a set to a certa degree, but t s probable that he s ot sure about t. I other words, there may be ucertaty about the membershp degree. he ma premse s that the parameters demad across the problem are ucerta. So, they are kow to fall a prescrbed ucertaty set wth some attrbuted degree. I Fuzzy Set (FS) theory, there s o meas to corporate ths hestato the membershp degree. o corporate the ucertaty the membershp degree, Itutostc Fuzzy Sets (IFSs) proposed by taassov [] s a exteso of FS theory. I IFS theory alog wth degree of membershp a degree of o-membershp s usually cosdered to express ll-kow quatty. hs degree of membershp ad omembershp fuctos are so defed as they are depedet to each other ad sum of them s less or equal to oe. So IFS s playg a mportat role decso makg uder ucertaty ad has gaed popularty recet years. However a applcato of the IFSs to optmzato problems troduced by gelov [3].Hs techque s based o maxmzg the degree of membershp, mmzg the degree of o-membershp ad the crsp model s formulated usg the IF aggregato operator. Now the fact s that IFS determate formato s partally lost, as hestat formato s take cosderato by default. So determate formato should be cosdered decso-makg process. Smaradache [4, 41-50] defed eutrosophc set that could hadle determate ad cosstet formato. I eutrosophc sets determacy s quatfed explctly as determacy membershp alog wth truth membershp, ad falsty membershp fucto whch are depedet.wag et.al [5] defe sgle valued eutrosophc set whch represets mprecse, complete, determate, cosstet formato. hus takg the uverse as a real le we ca develop the cocept of sgle valued eutrosophc set as specal case of eutrosophc sets. hs set s 7

74 Neutrosophc Operatoal Research Volume II able to express ll-kow quatty wth ucerta umercal value decso makg problem. It helps more adequately to represet stuatos where decso makers absta from expressg ther assessmets. I ths way eutrosophc set provdes a rcher tool to grasp mpresso ad ambguty tha the covetoal FS as well as IFSs. lthough exactly kow, partally ukow ad ucerta formato hadled by fully utlsg the propertes of trasto rate matrces, together wth the covexfcato of ucerta domas [6-8], NSO s more realstc applcato of optmum desg. hese characterstcs of eutrosophc set led to the exteso of optmzato methods Neutrosophc evromet (NSE). Besdes It has bee see that the curret research o fuzzy mathematcal programmg s lmted to the rage of lear programmg troduced by Zemmerma [9]. It has bee show that the solutos of Fuzzy Lear Programmg Problems (FLPPs) are always effcet. he most commo approach for solvg fuzzy lear programmg problem s to chage t to correspodg crsp lear programmg problem. But practcally there exst so may olear structural desgs such as welded beam desg problem varous the felds of egeerg. So developmet of olear programmg s also essetal. Recetly a robust ad relable statc output feedback (SOF) cotrol for olear systems [5] ad for cotuous-tme olear stochastc systems [6] wth actuator fault a descrptor system framework have bee studed. However weldg ca be defed as a process of og metallc parts by heatg to a sutable temperature wth or wthout the applcato of pressure. hs cost of weldg should be ecoomcal ad welded beam should be durable oe. Sce decades, determstc optmzato has bee wdely used practce for optmzg welded coecto desg. hese clude mathematcal tradtoal optmzato algorthms such as Davd-Fletcher-Powell wth a pealty fucto (DVID) [10], Grffth ad Stewart s Successve Lear pproxmato (PPROX) [10], Smplex Method wth Pealty Fucto (SIMPLEX) [10], Recherdso s Radom Method (RNDOM) [10], Harmoy Search Method [11], G based Method [1,13], Improved Harmoy Search lgorthm [14], Smple Costraed Partcle Swarm Optmzer (SC-PSO) [15], Mezura [16], Costraed Optmzato va PSO lgorthm (COPSO) [17], G based o a coevoluto model (G1) [18], G through the use of domace based touramet selecto (G) [19], Evolutoary Programmg wth a cultural algorthm (EP) [0], Co-evolutoary Partcle Swarm Optmzato (CPSO) [1], Hybrd Partcle swarm optmzato (HPSO) wth a feasblty based rule [], Hybrd Nelder- Mead Smplex search method ad partcle swarm optmzato (NM-PSO) [3], 73

75 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Partcle Swarm Optmzato (PSO) [4], Smulate elg (S) [4], Goldlke (GL) [4], Cuckoo Search (Cuckoo) [4], Frefly lgorthm (FF), Flower Pollato (FP) [4], t Lo Optmzer (LO) [4], Gravtatoal Search lgorthm (GS) [4], Mult-Verse Optmzer (MVO) [4] etc. ll these determstc optmzatos am to search the optmum soluto uder gve costrats wthout cosderato of ucertates. herefore, these tradtoal techques caot be applcable optmzg welded beam desg whe mprecseess s volved formato. hus, the research o optmzato for olear programmg uder fuzzy, IF ad eutrosophc evromet are ot oly ecessary the fuzzy optmzato theory but also has great ad wde value applcato to welded beam desg problem of coflctg ad mprecse ature. hs s the motvato of our preset vestgato. I ths regard, t ca be cted that Das et al. [7] developed eutrosophc olear programmg wth umercal example ad applcato of real lfe problem recetly. sgle obectve plae truss structure [8] ad a multobectve plae truss structure [9] have bee optmzed IF evromet. mult-obectve structural model has bee optmzed by IF mathematcal programmg wth IF umber for truss structure [30], welded beam structure [37] ad eutrosophc umber for truss desg [36] as coeffcet of obectve by Sarkaret.al. Wth the help of lear membershp [31] ad olear membershp [3, 33] for sgle obectve truss desg ad mult-obectve truss desg [34] have bee optmzed eutrosophc evromet. mult-obectve goal programmg techque [35] ad -orm, -coorm based IF optmzato techque [38] have bee developed to optmze cost of weldg eutrosophc ad IF evromet respectvely. he am of ths paper s to show the effcecy of sgle obectve NSO techque fdg optmum cost of weldg of welded beam mprecse evromet ad to make a comparso of results obtaed dfferet determstc methods. he paper s orgazed as follows. I sect., we have preseted mathematcal prelmares o eutrosophc set. I sect.3, we have developed mathematcal algorthm to solve a sgle obectve olear programmg problem. I sect. 4, we have studed detal formulato of welded beam ad solved t usg NSO techque. I sect. 5 we have solved welded beam desg model umercally. Lastly, sect.6 we arrve at a cocluso. 74

76 Neutrosophc Operatoal Research Volume II Mathematcal Prelmares I the followg, we brefly descrbe some basc cocepts ad basc operatoal laws related to eutrosophc set [1,, 5, 7].1 Fuzzy Set (FS) [1] Let X be a fxed set. fuzzy set of X s a obect havg the form (1) x x x X, where the fucto : 0,1 x X to the set. X stads for the truth membershp of the elemet.. Itutostc Fuzzy Set (IFS) [] Let a set X be fxed. tutostc fuzzy set or IFS of the form X s a obect,, () X x F x x X where : X 0,1 ad : 0,1 F X defe the truth membershp ad falsty membershp respectvely, for every elemet of x X such that 0 x F x Sgle-Valued Neutrosophc Set (SVNS) [5] Let a set X be the uverse of dscourse. sgle valued eutrosophc set over X s a obect havg the form,,, (3) x x I x F x x X : X 0,1, : 0,1 where I X ad : 0,1 F X are truth, determacy ad falsty membershp fuctos respectvely so as to 0 x I x F x 3 for all x X..4. Uo of Neutrosophc Sets (NSs) [7] he uo of two sgle valued eutrosophc sets valued eutrosophc set U deoted by ad B s a sgle U B x, x, I x, F x x X (4) U U U ad s defed by the followg codtos 75

77 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag () x x x max,, U B () I x I x I x max,, U B () F x m F x, F x for all x X U B for ype-i where Or aother way by defg followg codtos () x x x max,, U B () I x m I x, I x U B () F x m F x, F x for all x X U U B, x I U x, F U x for ype-ii represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be calculated for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.6, 0.4, 0. / x 0.5, 0., 0.3 / x 0.7, 0., 0. / x (5) ad for ype -II as 1 3 B 0.6, 0.1, 0. / x 0.5, 0., 0.3 / x 0.7, 0.1, 0. / x (6).5. Itersecto of Neutrosophc Sets 1 3 he tersecto of two sgle valued eutrosophc sets sgle valued eutrosophc set E s deoted by ad B s a E B x, x, I x, F x x X (7) E E E ad s defed by the followg codtos () x x x m,, E B () I x I x I x m,, E B 76

78 Neutrosophc Operatoal Research Volume II () F x max F x, F x for all x X E B for ype-i where Or, aother way by defg followg codtos () x x x m,, E B () I x max I x, I x E B () F x max F x, F x for all x X E E B x, I E x, F E x for ype-ii represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be measured for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.3, 0.1, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x (8) ad for ype -II as 1 3 B 0.3, 0.4, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0., 0.5 / x (9) Mathematcal alyss Decso makg s othg but a process of solvg the problem that acheves goals uder costrats. he outcome of the problem s a decso, whch should a acto. Decso-makg plays a mportat role dfferet subect such as feld of ecoomc ad busess, maagemet sceces, egeerg ad maufacturg, socal ad poltcal scece, bology ad medce, mltary, computer scece etc. It faces dffculty progress due to factors lke complete ad mprecse formato, whch ofte preset real lfe stuatos. I the decso makg process, the decso maker s ma target s to fd the value from the selected set wth the hghest degree of membershp the decso set ad these values support the goals uder costrats oly. However, there may arse stuatos where some values from selected set caot support, rather such values strogly agast the goals uder costrats, whch are o-admssble. I ths case, we fd such values from the selected set wth least degree of o- 77

79 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag membershp the decso sets. IFSs ca oly hadle complete formato ot the determate formato ad cosstet formato, whch exsts commoly belef system. I eutrosophc set, determacy s quatfed explctly ad truth-membershp, determacy-membershp ad falstymembershp are depedet to each other. herefore, t s atural to adopt for that purpose the value from the selected set wth hghest degree of truthmembershp, hghest degree or least degree of determacy-membershp ad least degree of falsty-membershp o the decso set. hese factors dcate that a decso makg process takes place eutrosophc evromet Neutrosophc Optmzato (NSO) echque to Solve Sgle obectve Nolear Programmg Problem (SONLPP) Nolear Programmg Problem (NLPP) may be cosdered the followg form Mmze g x (10) Subect to, ; 1,,..., m (11) g x b x 0 (1) Usually costrats goals are cosdered as fxed quatty. However, real lfe problems, the costrat goals caot be always exact. So we ca cosder the costrat goals for less tha type costrats at least to exted to b b for 1,,..., 0 b ad t may be possble m. hs fact seems to take the costrat goals as a eutrosophc set ad t wll be more realstc descrptos tha others. he NSO problem wth eutrosophc goals ca be descrbed as follows Mmze g x (13) Subect to,, 1,,..., m (14) g x b x 0 where represets equalty eutrosophc sese. (15) I the case of degree of falsty membershp ad determacy membershp t s to defe smultaeously wth degree of truth membershp of the obectve ad costrat ad whle all these three degrees are depedet of each other, NSO ca be used as a more geeral tool to descrbg ths ucertaty. Cosderg maxmzato of the degree of truth membershp together wth mmzato or maxmzato of the degree of determacy as per decso 78

80 Neutrosophc Operatoal Research Volume II maker s choce ad mmzg degree of falsty membershp of eutrosophc fuzzy obectve ad costrats, we ca formulate a NSO techque to solve a Neutrosophc Nolear Programmg(NSNLP)(Eq.13-Eq.15) problem. o solve the NSNLP (Eq.13-Eq.15), followg Warer s [40] ad gelov [39] we are presetg a soluto procedure by successve steps as follows Step-1: Followg Warer s approach solve the SONLPP wthout tolerace costrats (.e costrats (.e 0 Here they are, Sub-problem-1 g x b ),wth tolerace of truth membershp g x b b ) by approprate o-lear programmg techque Mmze g x (16) Subect to, ; 1,,..., m (17) g x b x 0 (18) Sub-problem- Mmze g x (19) Subect to, 0, 1,,..., m (0) g x b b x 0 (1) we may get optmal solutos x * x 1, g x * g x 1 ad x * x, g x * g x sub-problem 1 ad respectvely. for Step-: From the result of step 1 we ow fd the lower boud ad upper boud of obectve fuctos. Let U, U I, U F g x g x gx be the upper bouds of truth, determacy, falsty membershp fucto for the obectve respectvely ad L, L I, L F g x g x gx be the lower boud of truth, determacy, falsty membershp fuctos of obectve respectvely usg followg rules 1 max, g x U g x g x () 79

81 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag 1 m, g x L g x g x (3) But sgle valued NSO techque the degree of truth, determacy ad falsty membershp are cosdered so that the sum of these degree of membershp values are less tha three. o defe the falsty ad determacy membershp fucto of NLP(Eq.10-Eq.1),let us cosder upper ad lower boud of obectve fucto F U, g x g x F L g x such that ad U I g x, I L g x be the U F g x U g x (4) F gx gx gx gx 0 1 L L t U L where t (5) L I g x L g x (6) I gx gx gx gx 0 1 U L s U L where s (7) he tal eutrosophc model (Model -I) wth asprato levels of obectves ca be formulated as g x L g x Fd x (8) So as to satsfy wth tolerace U L for degree of truth membershp (9) g x gx g x L g x I I I wth tolerace U L for degree of determacy g x gx membershp (30) g x U g x F F F wth tolerace U L for degree of falsty membershp (31) g x gx wth tolerace g x b 0 b for degree of truth membershp (3) wth tolerace for degree of determacy membershp (33) g x g x b 0 g x b b wth tolerace 0 g x membershp (34) b b b For 1,,... m, t U L ; t 0,1 g x g x g x ; 0,1 ad g x g x g x s U L s for degree of falsty ad for Mode-II t ca be formulated as 80

82 Neutrosophc Operatoal Research Volume II g x L g x Fd x (35) So as to satsfy wth tolerace U L for degree of truth membershp (36) g x gx g x U g x I I I wth tolerace U L g x gx for degree of determacy membershp (37) g x U g x F F F wth tolerace U L for degree of falsty membershp (38) g x gx g x b wth tolerace 0 b wth tolerace g x b g x for degree of truth membershp (39) g x for degree of determacy membershp (40) 0 g x b b wth tolerace 0 g x b b b for degree of falsty membershp (41) for 1,,... m, t U L ; t 0,1 g x g x g x ; 0,1 ad g x g x g x s U L s Here ' ' deotes equalty eutrosophc sese. Step-3: Here for smplcty lear membershp determacy ad F g x are defed as follows g x for truth, I g x for for falsty membershp fuctos of obectve fucto 1 f g x Lg x U g x g x f L g x U 0 f g x U gx gx gx g x g x U L g x g x (4) I 1 f g x Lg x U g x I g x f L g x U I 0 f g x U gx I gx I I gx I I g x g x U L g x g x (43) 81

83 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag F 0 f g x Lg x g x L F g x f L g x U F 1 f g x U gx F gx F F gx F F g x g x U L g x g x Step-4: I ths step usg lear membershp fucto g x for truth, (44) I g x for determacy ad F g x membershp fuctos of costrats as follows for falsty membershp fuctos we ca calculate the 1 f g x b b b g x g g x f b g x b b x 0 0 f g x b b (45) 0 1 f g x b b g x g x I g x f b g x b g x g x g x f g x b g x 0 f g x b g x g x b F g x f b g x b b g x 0 g x 0 g x b g x 1 f g x b b 0 (47) (46) m b 0 for 1,,..., 0, g x g x. ad Step-5: Now usg NSO [31] for sgle obectve optmzato wth lear truth, determacy ad falsty membershp fuctos the NSNLP (Eq.13- Eq.15), ca be formulated as Model-I, gx g x Maxmze g x g x (48), gx g x Maxmze I g x I g x (49), gx g x Mmze F g x F g x (50) 8

84 Neutrosophc Operatoal Research Volume II Such that x I x F x (51) 3; g x g x g x 3; (5) x I x F x g g g x I x ; g x gx x F x (53) ; g x g gx ; (54) x I x g g ; (55) x F x g x I x F x,, 0,1 g x g x g x x, I x, F x [0,1] g g g x 0 1,,..., m (56) Model-II, gx g x Maxmze g x g x (57), gx g x Mmze I g x I g x (58), gx g x Mmze F g x F g x (59) subect to the same costrats as Model-I ll these crsp olear programmg problems (Model -I), (Model-II) ca be solved by approprate mathematcal algorthm. 4 Welded Beam Desg (WBD) ad ts Optmzato Neutrosophc Evromet Weldg, a process of og metallc parts wth the applcato of heat or pressure or the both, wth or wthout added materal, s a ecoomcal ad effcet method for obtag permaet ots the metallc parts. hese welded ots are geerally used as a substtute for rveted ot or ca be used as a alteratve method for castg or forgg. he weldg processes ca broadly be classfed to followg two groups, the weldg process that uses heat aloe to o two metallc parts ad the weldg process that uses a combato of heat 83

85 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag ad pressure for og (Bhadar. V. B). However, above all the desg of welded beam should preferably be ecoomcal ad durable oe. 4.1 WBD Formulato he optmum welded beam desg(fg. 1) ca be formulated cosderg some desg crtera such as cost of weldg.e cost fucto, shear stress, bedg stress ad deflecto,derved as follows Cost Fucto Formulato he performace dex approprate to ths desg s the cost of weld assembly. he maor cost compoets of such a assembly are () set up labour cost, () weldg labour cost, () materal cost,.e where, C X C0 C1 C (60) C X materal cost. Now Set up cost cost fucto; C 0 C0 set up cost; C1 weldg labour cost; C : he compay has chose to make ths compoet a weldmet, because of the exstece of a weldg assembly le. Furthermore, assume that fxtures for set up ad holdg of the bar durg weldg are readly avalable. he cost ca therefore be gored ths partcular total cost model. C 0 Weldg labour cost C 1 : ssume that the weldg wll be doe by mache at a total cost of $10/hr (cludg operatg ad mateace expese). Furthermore suppose that the mache ca lay dow a cubc ch of weld 6 m. he labour cost s the $ 1 $ m $ C V 1 3 w V 3 hr 60 m w (61) Where Vw weld volume, 3 Materal cost C : C C3V w C4VB (6) Where C3 cost per volume per weld materal,$/ 3 (0.37)(0.83) ; cost per volume of bar stock,$/ 3 (0.37)(0.83) ; VB volume of bar, 3. B C4 From geometry V h l ;volume of the weld materal, 3 ; V x x ad w ; volume of bar, 3 V x x L x V tb L l bar herefore cost fucto become C X h l C h l C tb L l x x x x x (63) weld 1 84

86 Neutrosophc Operatoal Research Volume II Costrats Dervato from Egeerg Relatoshp Fg. 1 Shear stresses the weld group. Maxmum shear stress weld group o complete the model t s ecessary to defe mportat stress states Drect or prmary shear stress.e Load P P P 1 (64) hroat area hl x x Sce the shear stress produced due to turg momet M P. e at ay secto, s proportoal to ts radal dstace from cetre of gravty of the ot G, therefore stress due to M s proportoal to ad s a drecto at rght agles to. I other words costat (65) R r R 1 R herefore x x x l h t R (66) Where, s the shear stress at the maxmum dstace R ad s the shear stress at ay dstace r. Cosder a small secto of the weld havg area d a dstace from G. herefore shear force o ths small secto turg momet of the shear force about cetre of gravty s r dm d r d r R at d ad (67) herefore total turg momet over the whole weld area M d r J. R (68) R 85

87 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag where J polar momet of erta of the weld group about cetre of gravty. herefore shear stress due to the turg momet.e. secodary shear stress, MR J (69) I order to fd the resultat stress, the prmary ad secodary shear stresses are combed vectorally. herefore the maxmum resultat shear stress that wll be produced at the weld group, cos, (70) 1 1 where, agle betwee 1 ad. s l x R R (71) cos ; x 1 1. (7) R Now the polar momet of erta of the throat area gravty s obtaed by parallel axs theorem, about the cetre of x x l l x 1 3 J I xx x x x x1x (73) where, throat area x1x, l Legth of the weld, x Perpedcular dstace betwee two parallel axes Maxmum bedg stress beam t h x x 1 3 (74) Now Maxmum bedg momet where PL; PL, maxmum bedg stress Z, I Z secto modulus ; y I momet of erta 3 bt ; 1 t extreme fbre from cetre of gravty of cross secto ; herefore y dstace of bt Z. 6 6PL 6PL Z bt x x So bar bedg stress x. (75) Maxmum deflecto beam Maxmum deflecto at catlever tp

88 Neutrosophc Operatoal Research Volume II PL PL 4PL 4PL x 3 3 3EI bt Ebt Ex4x (76) 3 3E 1 Bucklg load of beam l Bucklg load ca be approxmated by PC x EIC a El 1 l C (77) 6 tb E 36 L t E EGx3x 4 /36 x3 E 1 L 4 G 1 L L 4 G (78) where, I momet of erta 4. Crsp Formulato of WBD 3 bt 1 3 ; torsoal rgdty C GJ tb G; 1 3 t l L; a. I desg formulato a welded beam ([10],Fg. ) has to be desged at mmum cost whose costrats are shear stress weld,bedg stress the beam,bucklg load o the bar P,ad deflecto of the beam.he x1 h x l desg varables are where h s the weld sze, l s the legth of the weld, x 3 t x4 b s the depth of the welded beam, b s the wdth of the welded beam. t Fg. Desg of the welded beam 87

89 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag he sgle-obectve crsp welded beam optmzato problem ca be formulated as follows (79) Mmze C X x x x x x such that (80) g x x 1 max 0 (81) g x x max 0 (8) g x x x g x x x x x 14 x 5 0 (83) (84) g x x 5 1 (85) g x x 6 max 0 (86) g x P P x 7 C x, x, x, x 0,1 (87) x R where x R x x1 x3 1 1 ; 1 4 ; x1 x x x1 x3 J ; 1 P MR ; x ; M P L xx J ; 1 6PL xx x ; x PL ; Ex x EGx3x 4 / 36 x3 E PC x 1 as derved as Eq.(70), Eq.(64), Eq.(69), L L 4 G Eq.(68), Eq.(66), Eq.(73), Eq.(75),, Eq.(76),, Eq.(78) respectvely. ga P Force o beam ; L Beam legth beyod weld; Heght of the welded beam; x Legth of the welded beam; x3 Depth of the welded beam; x4 Wdth of the welded beam; x Desg shear stress; x Desg ormal stress for beam materal; M Momet of P about the cetre of gravty of the weld, x Polar momet of erta of weld group; G Shearg modulus of Beam Materal; E Youg modulus; max 1 Desg Stress of the weld; max 4 3 J Desg ormal stress for the beam materal; max Maxmum deflecto; 1 Prmary stress o weld throat, Secodary torsoal stress o weld. 88

90 Neutrosophc Operatoal Research Volume II 4.3 WBD Formulato Neutrosophc Evromet Sometmes slght chage of stress or deflecto ehaces the weght of structures ad drectly cost of processg. I such stuato whe decso maker (DM) s doubt to decde the stress costrat goal, the DM ca duce the dea of acceptace boudary, hestacy respose or egatve respose marg of costrats goal. hs fact seems to take the costrat goal as a NS stead of FS ad IFS. It may be more realstc descrpto tha FS ad IFS. Whe the sheer stress, ormal stress ad deflecto costrat goals are NS ature the above crsp welded beam desg (Eq.79-Eq.87) ca be formulated as (88) Mmze C X x x x x x Such that (89) g x x 1 max (90) g x x max (91) g x x x g x x x x x 14 x 5 0 (9) (93) g x x 5 1 (94) g x x 6 max (95) g x P P x 7 C x, x, x, x 0,1 (96) Where all the parameters have ther usual meag as stated sect.4..here costrat goals are characterzed by Neutrosophc Sets max max x1, x, max x1, x, I max x1, x, F max x1, x (97) max max max wth max x1, x, I max x1, x, F max x1, x max max as the degree of truth, determacy ad falsty membershp fucto of Neutrosophc set ; max max 3 4 max 3 4 max 3 4 max 3 4 max max max max max x, x, x, x, I x, x, F x, x (98) wth max x3, x4, I max x3, x4, F max x3, x4 max max as the degree of truth, determacy ad falsty membershp fucto of Neutrosophc set ad max max 89

91 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag max max 3 4 max 3 4 max 3 4 max 3 4 max max max x, x, x, x, I x, x, F x, x (99) wth max x3, x4, I max x3, x4, F max x3, x4 max max as the degree of truth, determacy ad falsty membershp fucto of Neutrosophc set 4.4 Optmzato of WBD Neutrosophc Evromet max max o solve the WBD (Eq.88-Eq.96)), step 1 of sect.3.1 s used ad we wll get optmum solutos of two sub problem as ad. fter that accordg to step we fd upper ad lower boud of membershp fucto of obectve fucto as U, U I, U F C X C X CX ad L, L I, L F C X C X CX where U C X C X 1 1 max,, L C C X X C X C X U U L L X 1 X m,, herefore, where 0 U C X C X L (100) C X F F C X C X C X C X C X L L U L, where 0 U C X C X L (101) C X I I C X C X C X C X C X Let the lear membershp fuctos for obectve be, C X 1 f C X L U C X C X f L C X U C X 0 f C X UC X C X C X C X C X U L C X W 1 f C X L L C X C X C X I C X C X f L C X L C X C X C X C X 0 f W LC X C X L C X C X 0 f C X C X L C X C X F C X f L C X U 1 f C X UC X C X C X C X C X U L C X C X C X (10) (103) (104) ad costrats be, 90

92 Neutrosophc Operatoal Research Volume II 1 f X 0 X X f X X 0 0 f X f X X X I X X f X X 0 f X X X X 0 f X X F X f X X 0 1 f X X 0 0 X X (105) (107) (106) for 1,,..., m 0, 0 X X he NSO problem (Eq.88-Eq.96)), ca be formulated as the followg crsp lear programmg problem by cosderg lear membershp as follows, ype-i Maxmze (108) Such that ; C X C X (109) C X C X U L U ; C X C X C X C X C X C X U L L (110) ; C X C X C X C X C X C X U L L (111) ; X U L U (11) X X X ; X U L U (113) X X X X X ; X U L L (114) X X X X X 3; ; ;,, 0,1 91

93 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Model-II Maxmze (115) Such that ; C X C X (116) C X C X U L U U ; C X (117) C X C X ; C X C X C X (118) C X C X U L L ; X U L U (119) X X X X U ; (10) X X ; X U L L (11) X X X X 3; ; ; (1),, 0,1 ll these crsp olear programmg problem ca be solved by approprate mathematcal algorthm. 5 Numercal Illustrato Iput data of welded beam desg problem(eq.79-eq.87) able 1as follows able 1 : Iput data for eutrosophc model (Eqs.(88-96)) are gve ppled load P lb Beam legth beyod weld L Youg Modulus E ps Value of G ps Maxmum allowable shear stress max ps wth allowable tolerace 50 Maxmum allowable ormal stress max ps wth allowable tolerace 50 Maxmum allowable deflecto max 0.5 wth allowable tolerace

94 Neutrosophc Operatoal Research Volume II Soluto: ccordg to step of sect. 3.1, we fd upper ad lower boud of membershp fucto of obectve fucto as U, U I, U F C X C X CX ad U U, L L,, I, F F I L L L C X C X CX where C X C X CX CX L I C X,wth C X adu L C X C X C X wth F C X C X ; Now usg the bouds we calculate the membershp fuctos for Model- I obectve as follows 1 f C X C X C X f C X C X f C X (13) 1 f C X gx C X I C X f C X C X C X 0 f C X C X C X C X 0 f C X C X C X F C X C X f C X C X C X 1 f C X (14) (15) smlarly the membershp fuctos for shear stress costrat are, x 1 f g g1 x g 1 1 x f g1 x g x 50 0 f g1 x (16) x 1 f g g x I g x f g x 0 f g1 x g1 x g x g1 x g1 x g1 x (17) 93

95 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag x 0 f g g1 x g1 x g1 x F g 1 1 x g x f g x g x 50 g1 x 1 f g1 x (18) where 0 1, g x g x ad the membershp fuctos for ormal stress costrat are, x 1 f g g x g x f g x g x 50 0 f g x x 1 f g g x I g x f g x 0 f g x g x g x gx g x g x (19) (130) x 0 f g g x g x g x F g x g x f g x g x 50 g x 1 f g x (131) where 0, 50 g x g x ad the membershp fuctos for deflecto costrat are, x 1 f g g6 x g 6 6 x f 0.5 g6 x 0.3 g x f g6 x 0.3 x 1 f g g x I g x f g x 0 f g6 x 0.5 g6 x g x g6x g6 x g6 x (13) (133) 94

96 Neutrosophc Operatoal Research Volume II x 0 f g6 0.5 g6 x g6 x 0.5 g6 x F g 6 6 x g x f g x g x 0.05 g6 x 1 f g6 x 0.3 (134) where 0 6, 6.05 g x g x Smlarly the truth,determacy ad falsty membershp fucto ca be calculated for obectve ad costrat fuctos.now, usg above metoed truth, determacy ad falsty lear membershp fuctos NLP (Eq.79-Eq.87)) ca be solved for Model -I, Model-II, by fuzzy, tutostc fuzzy ad NSO techque wth dfferet values of ad,,,,, 1, C X g x g x g6 x C X g1 x g x g6 x. he optmum heght, legth, depth, wdth ad cost of weldg of welded beam desg (Eq.79-Eq.87) are gve able ad the soluto are compared wth other determstc optmzato methods. able : Comparso of Optmal soluto of welded beam desg (Eq.79- Eq.87)) based o fuzzy ad IF ad NSO techque (Model - I ad Model- II) wth dfferet methods Methods Heght DVID[10] PPROX[10] SIMPLEX[10] RNDOM[10] Harmoy Search lgorthm[11] G Based Method[1] G Based Method[13] Improved Harmoy Search lgorthm[14] x 1 ch Legth x ch Depth x 3 ch Wdth x 4 ch Weldg cost SC-PSO[15] Mezura [16] COPSO[17] G1[18] G[19] EP[0] CPSO[1] HPSO[] C X $ 95

97 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag NM-PSO[3] PSO[4] S[4] GL[4] Cuckoo [4] FF[4] FP[4] LO[4] GS[4] MVO[4] Fuzzy sgle-obectve o-lear programmg [8] Itutostc Fuzzy sgle-obectve olear programmg (FSONLP) [8] CX.0015, g x 5, g g 6 Proposed Neutosophc optmzato(nso) CX.0015, g1 x g x 5, g6 x CX.004, g1 x g x 40, g6 x x 1 x 5,.05, 5,.05, 40,.04, Model-I Model-II detaled comparso has bee made amog several determstc optmzato methods for optmzg weldg cost wth mprecse optmzato methods such as fuzzy, IF ad NSO methods able. It has bee observed that fuzzy olear optmzato provdes better result comparso wth IF ad NSO methods. lthough t has bee see that cost of weldg s mmum other tha the method studed ths paper, as far as o-determstc optmzato methods cocer,fuzzy, IF ad NSO are provdg a valuable result mprecse evromet ths paper ad lterature. It has bee see that Improved Harmoy Search lgorthm[14],copso[17],ep[0],hpso[] are provdg mmum most cost of weldg where all the parameters have bee cosdered as exact ature. However, t may also be oted that the effcecy of the proposed method depeds o the model chose to a greater extet because t s ot always expected that NSO wll provde better results over fuzzy ad IF optmzato. So overall NSO s a effcet method fdg best optmal soluto mprecse evromet. It has bee studed that same results have bee obtaed 96

98 Neutrosophc Operatoal Research Volume II whle determat membershp tred to be maxmze (Model- I) or mmze (Model-II) NSO for ths partcular problem. 6 Coclusos ad Future Work I ths paper, a sgle obectve NSO algorthm has bee developed by defg truth, determacy ad falsty membershp fucto, whch are depedet to each other. Usg ths method frstly optmum heght, legth, depth, wdth ad cost of weldg have bee calculated ad fally the results are compared wth dfferet determstc methods. So llustrated example of welded beam desg has bee provded to llustrate the optmzato procedure, effectveess ad advatages of the proposed NSO method. he comparso of NSO techque wth other optmzato techques has ehaced the acceptablty of proposed method.he proposed procedures has ot oly valdated by the exstg methods but also t develops a ew drecto of optmzato theory mprecse evromet, whch s more realstc. ckowledgemets he research work of Mrdula Sarkar s faced by Rav Gadh Natoal Fellowshp (F1-17.1/ SC-wes-4549/(S-III/Webste)), Govt of Ida. Refereces [1] Zadeh, L.. Fuzzy set. Iformato ad Cotrol.1965:8: [] taassov, K.. Itutostc fuzzy sets. Fuzzy Sets ad Systems. 1986:0 : [3] gelov,p.p. Optmzato Itutostc Fuzzy Evromet, Fuzzy Sets ad Systems. 1997:86: [4] Wag, H., Smaradache, F., Zhag, Y. Q., Rama,R. Sgle valued eutrosophc sets, Multspace ad Multstructure. 010:4: [5] We Y. Qu, J., Karm, H. R., Wag,Mao.: H model reducto for cotuoustme Markova ump systems wth complete statstcs of mode formato. Iteratoal Joural of Systems Scece.014: 45 : [6] We,Y. Qu, J., Karm, H. R., Wag,Mao.New results o H1 dyamc output feedback cotrol for Markova ump systems wth tme-varyg delay ad defectve mode formato. Optm. Cotrol ppl. Meth. 014:35: [7] We,Y. Qu, J., Karm, H. R., Wag, Mao. Flterg desg for two-dmesoal Markova ump systems wth state-delays ad defcet mode formato. Iformato Sceces.014:69: [8] Zmmerma, H.J.Fuzzy lear programmg wth several obectve fucto. Fuzzy sets ad systems.1978:1: [9] Ragsdell, K.M., Phllps, D.. Optmal desg of a class of welded structures usg geometrc programmg, SME Joural of Egeerg for Idustres, 1976:98 : [10] Lee,K.S., Geem,Z.W. ew meta-heurstc algorthm for cotuous egeerg optmzato: harmoy search theory ad practce.comput. Methods ppl. Mech. Egrg. 005:194 :

99 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [11] Deb, K., Pratap,., Motra, S.Mechacal compoet desg for multple obectves usg eltst o-domated sortg G. I Proceedgs of the Parallel Problem Solvg from Nature VI Coferece, Pars.000:16 0, [1] Coello, C..C. Use of a self-adaptve pealty approach for egeerg optmzato problems. Comput. Id.000b:41: DOI: /S (99) [13] Mahdav, M., Fesaghary, M., Damagr, E. mproved harmoy search algorthm for solvg optmzato problems. ppled Mathematcs ad Computato. 007:188 : [14] Carlos. Coello Coello, Solvg Egeerg Optmzato Problems wth the Smple Costraed Partcle Swarm Optmzer, Iformatca.008:3: [15] Coello, C..C., Motes, E.M. Useful feasble solutos egeerg optmzato wth evolutoary algorthms. Mexca Iteratoal Coferece o rtfcal Itellgece.005: DOI: / _66. [16] gurre,.h.,zavala,.m.,dharce,e.v.,roda,s.b.copso:costraed Optmzato va PSO lgorthm.cetre for Research Mathematcs(CIM).007:echcal report No.I-07-04/ [17] Coello Coello,C.. Use of a self-adaptve pealty approach for egeerg optmzato problems.computers Idustry.000:41: [18] Coello Coello, C.., Motes, E.M. Cotrat-hadlg techques geetc algorthms through domace based touramet selecto.dvaced Egeerg Iformatcs.00:16: [19] Coello Coello, C.., Becerra, R.L., Effcet evolutoary optmzato through the use of a cultural algorthm. Egeerg Optmzato.004:36: [0] He, Q. Wag, L. effectve co-evolutoary partcle swarm optmzato for costraed egeerg desg problems.egeerg pplcatos of rtfcal Itellgece.007a:0: [1] He, Q. Wag, L. hybrd partcle swarm optmzato wth a feasblty based rule for costraed optmzato.ppled Mathematcs ad Computato. 007b:186: [] Zahara,E., Kao,Y..Hybrd Nelder-mead smplex search ad partcle swarm optmzato for costraed egeerg desg problems.expert System wth pplcatos.009:36: [3] Davd, D.C.N, Stephe, S.E., oy.j.. Cost mmzato of welded beam desg problem usg PSO, S, PS, GOLDLIKE, CUCKOO, FF, FP, LO, GS ad MVO.Iteratoal Joural of ppled Mathematcs.016:5:1-14. [4] We, Y., Qu, J., Karm, H. R. Relable Output Feedback Cotrol of Dscrete- me Fuzzy ffe Systems Wth ctuator Faults.016: Do: /CSI :1-1. [5] We,Y., Qu,J., Lam,H.K., Wu,L., pproaches to -S Fuzzy-ffe-Model- Based Relable Output Feedback Cotrol for Nolear Itˆo Stochastc Systems.016 :DOI /FUZZ : [6] Das, P., Roy,.K.,Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem. Neutrosophc Sets ad Systems. 015:9: [7] Sgh,B.; Sarkar,M.,Roy,. K.Itutostc Fuzzy Optmzato of russ Desg: Comperatve Study.Iteratoal Joural of Computer &Orgazato reds (IJCO).016:3:5-33. [8] Sarkar,M., Roy,. K. Itutostc Fuzzy Optmzato o Structural Desg: Comparatve Study.Iteratoal Joural of Iovatve Research Scece,Egeerg ad echology.016:5:

100 Neutrosophc Operatoal Research Volume II [9] Sarkar,M., Roy,. K.russ Desg Optmzato wth Imprecse Load ad Stress Itutostc Fuzzy Evromet.Joural of Ultra Scetst of Physcal Sceces.017:9:1-3. [30] Sarkar,M. Dey, S.,Roy,. K.russ Desg Optmzato usg Neutrosophc Optmzato echque.neutrosphc Sets ad Systems.016:13:6-69. [31] Sarkar,M., Roy,. K.russ Desg Optmzato usg Neutrosophc Optmzato echque: Comparatve Study.dvaces Fuzzy Mathematcs.017:1: [3] Sarkar,M.; Dey, S.,Roy,. K.Neutrosophc optmzato techque ad ts applcato o structural desg.joural of Ultra Scetst of Physcal Sceces.016:8: [33] Sarkar,M.; Dey, S., Roy,. K.Mult-Obectve Neutrosophc Optmzato echque ad ts pplcato to Structural Desg.Iteratoal Joural of Computer pplcatos,016:148: [34] Sarkar,M.,Roy,. K.Mult-Obectve Welded Beam Optmzato usg Neutrosophc Goal Programmg echque. dvaces Fuzzy Mathematcs.017:1: [35] Sarkar,M., Roy,. K.:russ Desg Optmzato wth Imprecse Load ad Stress Neutrosophc Evromet. dvaces Fuzzy Mathematcs.017:1: [36] Sarkar,M., Roy,. K.Optmzato of Welded Beam wth Imprecse Load ad Stress by Parameterzed Itutostc Fuzzy Optmzato echque. dvaces Fuzzy Mathematcs.017:1: [37] Sarkar,M., Roy,. K.Mult-Obectve Welded Beam Desg Optmzato usg -Norm ad -Coorm based Itutostc Fuzzy Optmzato echque.dvaces Fuzzy Mathematcs.017:1: [38] gelov,p.p.: Optmzato Itutostc Fuzzy Evromet, Fuzzy Sets ad Systems.1997:86: [39] Werer, B. Iteractve Fuzzy Programmg Systems,Fuzzy Sets ad Systems.1987:3: [40] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [41] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [4] bdel-basset, Mohamed, Ma Mohamed, ad ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [43] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [44] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [45] bdel-baset, Mohamed, Ibrahm M. Hezam, ad Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [46] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [47] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016): [48] Mrdula Sarkar, Samr Dey, apa Kumar Roy: russ Desg Optmzato usg Neutrosophc Optmzato echque, Neutrosophc Sets ad Systems, vol. 13, 016, pp do.org/10.581/zeodo

101 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [49] Ptu Das, apa Kumar Roy: Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem, Neutrosophc Sets ad Systems, vol. 9, 015, pp do.org/10.581/zeodo [50] Sabu Sebasta, Floret Smaradache: Exteso of Crsp Fuctos o Neutrosophc Sets, Neutrosophc Sets ad Systems, vol. 17, 017, pp [51] Ema.M.El-Nakeeb, Hewayda ElGhawalby,.. Salama, S..El-Hafeez: Neutrosophc Crsp Mathematcal Morphology, Neutrosophc Sets ad Systems, Vol. 16 (017), pp do.org/10.581/zeodo [5].. Salama, Hewayda ElGhawalby, Shmaa Fath l: opologcal Mafold Space va Neutrosophc Crsp Set heory, Neutrosophc Sets ad Systems, vol. 15, 017, pp do.org/10.581/zeodo

102 Neutrosophc Operatoal Research Volume II V Mult-Obectve Neutrosophc Optmzato echque ad Its pplcato to Rser Desg Problem Mrdula Sarkar 1 Ptu Das apa Kumar Roy 3 1 Ida Isttute of Egeerg Scece ad echology,shbpur, P.O. Botac Garde, Howrah, , Ida. Emal: mrdula.sarkar86@redffmal.com Staada College, Nadgram, Purba Medpur, 71631,West Begal, Ida. Emal: meptudas@yahoo.com 3 Ida Isttute of Egeerg Scece ad echology,shbpur, P.O. Botac Garde, Howrah, , Ida. Emal: roy_t_k@yahoo.co. bstract hs chapter ams to gve computatoal algorthm to solve a multobectve o-lear programmg problem (MONLPP) usg eutrosophc optmzato method. he proposed method s for solvg MONLPP wth sgle valued eutrosophc data. comparatve study of optmal soluto has bee made betwee tutostc fuzzy ad eutrosophc optmzato techque. he developed algorthm has bee llustrated by a umercal example. Fally optmal rser desg problem s preseted as a applcato of such techque. Keywords Neutrosophc Set, Sgle Valued Neutrosophc Set, Neutrosophc Optmzato, Mult-Obectve Rser Desg optmzato. 1 Itroducto he cocepts of Fuzzy Sets were troduced by Zadeh 1965[1]. Sce the Fuzzy Sets ad Fuzzy Logc have bee appled may real applcatos to hadle ucertaty. he tradtoal fuzzy sets uses oe real value x0,1 to represet the ruth membershp fucto of FS defed o uverse X.Sometmes x tself s ucerta ad hard to be defed by a crsp value. So the cocept of terval valued fuzzy sets was proposed [] to capture the ucertaty of truth membershp. I some applcatos we should cosder ot 101

103 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag oly the truth membershp supported by the evdet but also the falsty membershp agast by the evdet. hat s beyod the scope of fuzzy sets ad terval valued fuzzy sets. I 1986, taassov troduced the tutostc Fuzzy Sets [3], [5], whch s a geeralzato of FS. he IFS cosder both ruth membershp ad Falsty membershp. Itutostc Fuzzy set ca oly hadle complete formato ot the determate formato ad cosstet formato. I eutrosophc sets determacy s quatfed explctly ad truth, determacy ad falsty membershp fuctos are depedet. Neutrosophc Set was frst troduced by Smaradache 1995 [4, 16-5]. he motvato of the preset study s to gve computatoal algorthm for mult-obectve o-lear programmg problem by sgle valued eutrosophc optmzato approach. We also am to study the mpact of truth, determacy ad falsty membershp fuctos such optmzato process ad thus have made comparatve study tutostc fuzzy ad eutrosophc optmzato techque. lso as a applcato of such optmzato techque optmal rser desg problem s preseted. Mathematcal Prelmares I the followg, we brefly descrbe some basc cocepts ad basc operatoal laws related to eutrosophc set.1 Fuzzy Set(FS) Let X be a fxed set. fuzzy set of X s a obect havg the form x, x x X where the fucto : 0,1 (1) x X to the set... Itutostc Fuzzy Set(IFS) X stads for the truth membershp of the elemet Let a set X be fxed. tutostc fuzzy set or IFS of the form X x F x x X where : X 0,1 ad : 0,1 X s a obect,, () F X defe the truth membershp ad falsty membershp respectvely, for every elemet of x X such that 0 x F x 1. 10

104 Neutrosophc Operatoal Research Volume II.3. Sgle-Valued Neutrosophc Set(SVNS) Let a set X be the uverse of dscourse. sgle valued eutrosophc set over X s a obect havg the form,,, (3) x x I x F x x X : X 0,1, : 0,1 where I X ad : 0,1 F X are truth, determacy ad falsty membershp fuctos respectvely so as to x I x F x for all x X..4. Uo of Neutrosophc Sets(NSs) he uo of two sgle valued eutrosophc sets valued eutrosophc set deoted by U 0 3 ad B s a sgle U B x, x, I x, F x x X (4) U U U ad s defed by the followg codtos () x x x max,, U B () I x I x I x max,, U B () F x m F x, F x for all x X U B for ype-i Or aother way by defg followg codtos () x x x max,, U B () I x m I x, I x U B () F x m F x, F x for all x X where, U U B x, U for ype-ii U I x F x represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be calculated for 1 3 ad B s a sgle valued eutrosophc 103

105 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag ype -I as B 0.6, 0.4, 0. / x 0.5, 0., 0.3 / x 0.7, 0., 0. / x (5) ad for ype -II as 1 3 B 0.6, 0.1, 0. / x 0.5, 0., 0.3 / x 0.7, 0.1, 0. / x (6).5. Itersecto of Neutrosophc Sets 1 3 he tersecto of two sgle valued eutrosophc sets sgle valued eutrosophc set E s deoted by ad B s a E B x, x, I x, F x x X (7) E E E ad s defed by the followg codtos () x x x m,, E B () I x I x I x m,, E B () F x max F x, F x for all x X E B Or aother way by defg followg codtos () x x x m,, E B () I x max I x, I x E B () F x max F x, F x for all x X where, E E B x, E for ype-i for ype-ii E I x F x represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be measured for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.3, 0.1, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x (8) ad for ype -II as 1 3 B 0.3, 0.4, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0., 0.5 / x (9)

106 Neutrosophc Operatoal Research Volume II 3 Neutrosophc Optmzato echque to solve Mmzato ype Mult-Obectve No-lear Programmg Problem (P1) o-lear mult-obectve optmzato problem s of the form Mmze f1 x, f x,..., f p x (10) g x b, 1,,..., q (11) Now the decso set D, a coucto of eutrosophc obectves ad p q costrats s defed as D G C x, x, I x, F x Here k D D D k1 1 (1) m,,..., ;,,..., (13) x x x x x x x for all x X D G1 G Gp C1 C Cp m,,..., ;,,..., (14) I x I x I x I x I x I x I x for all x X D G1 G Gp C1 C Cp m,,..., ;,,..., (15) F x F x F x F x F x F x F x for all x X D G1 G Gp C1 C Cp x I x F x are ruth membershp fucto, Idetermacy where,, D D D membershp fucto, Falsty membershp fucto of Neutrosophc decso set respectvely. Now usg the eutrosophc optmzato the problem (P) s trasferred to the olear programmg problem as (P) Maxmze (16) Mmze (17) Maxmze (18) Such that x (19) G k x (0) C I x (1) G 105

107 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag C I x G k F x C F x () (3) (4) 3; (5), (6) (7),, 0,1 (8) Now ths o-lear programmg problem (P) ca be easly solved by approprate mathematcal algorthm to gve soluto of mult-obectve lear programmg problem (P1) by eutrosophc optmzato approach. Computatoal lgorthm Step-1 Solve the MONLP(1) as a sgle obectve o-lear problem p tmes for each problem by takg oe of the obectves at a tme ad gorg the others.hese soluto are kow as deal solutos. Let be the respectve optmal soluto for the k th dfferet obectve ad evaluate each obectve values for all these k th optmal soluto. Step- From the result of Step-1,determe the corresepodg values for every obectve for each derved soluto.wth the values of all obectves at each deal solutos,pay-off matrx ca be formulated as follows U as k Step-3 For each obectve U f x... p *... p f x f x f x f1 x f x f x f1 x f x f x * p p * p... p fk k x x, the lower boud L ad upper boud k r* k max (9) k r* ad Lk m fk x (30) where r 1,,..., k for truth membershp fucto of obectves. 106

108 Neutrosophc Operatoal Research Volume II Step-4 We represets upper ad lower bouds for determacy ad falsty membershp of obectves as follows U U F k I k U k ad L F k Lk Uk Lk (31) U ad L I k k Lk Uk Lk (3) Here t ad s are predetermed real umber 0,1 Step-5 Defe ruth membershp, Idermacy membershp,falsty membershp fuctos as follows 1 f fk x Lk U f x f x f L f x U k k k k k k k Uk Lk 0 f fk x U k I 1 f fk x Lk U f x I f x f L f x U I k k I I k k k I I k k Uk Lk I 0 f fk x U k F 1 f fk x Lk f x L F f x f L f x U F k k F F k k k F F k k Uk Lk F 0 f fk x U k (33) (34) (35) Step-6 Now eutrosophc optmzato method for MONLP problem gves a equvalet olear-programg problem as (P3) Max Such that k (36) k ; f x (37) k k ; I f x (38) k k ; F f x (39) 3; (40) 107

109 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag ; ; (41) (4),, 0,1 ; (43) ; 0 (44) g x b x k 1,,.., p; 1,,.., q (45) Whch s reduced to equvalet o-lear programmg problem as (P4) Max Such that (46). f x U L U (47) k k k k. f x U L U (48) I I I k k k k. f x U L L (49) F F F k k k k 3; (50) ; ; (51) (5),, 0,1 ; (53) ; 0 (54) g x b x k 1,,.., p; 1,,.., q Mathematcal Prelmares 4 Illustrated Example (P5) 1, (55) M f x x x x , (56) M f x x x x Such that x x (57) x1, x 0 (58) 108

110 Neutrosophc Operatoal Research Volume II Soluto: I L1 L1 6.75, Here L1 6.75, I adu s, F U1 U F ad t, L 57.87, F L L F I I L t, L L 57.87, adu s s 0.4.We take L ad t 0.3 ad able 1 : Comparso of optmal soluto by IFO ad NSO echque Optmzato echque Itutostc Fuzzy Optmzato(IFO) Proposed Neutrosophc Optmzato (NSO) Optmal Decso Varables x, x * * Optmal Obectve Fuctos f, f * * sprato levels of ruth, Falsty ad Idetermacy Membershp Fuctos * * * * * Sum of Optmal Obectve Values able 1 shows that eutrosophc optmzato techque gves better result tha Itutostc Fuzzy Nolear Programmg echque. 5 pplcato of Neutrosophc Optmzato Rser Desg Problem he fucto of a rser s to supply addtoal molte metal to a castg to esure a shrkage porosty free castg.shrkage porosty occurs because of the crease desty from the lqud to sold state of metals.o be effectve a rser must be soldfy after castg ad cota suffcet metal to feed the castg. Castg soldfcato tme s predcted from Chvorov s rule. Chvorov s rule provdes gudace o why rsers are typcally cyldrcal.he logest soldfcato tme for a gve volume s the oe where the shape of the part has the mmum surface area. From a practcal stadpot cylder has least surface area for ts volume ad easest to make. Sce the rser should soldfy 3 cylder sde rser whch cossts of a cylder of heght H ad dameter D. he theoretcal bass for rser desg s Chvorov s rule whch s / t k V S (59) where t soldfcato tme (mutes/secods), K soldfcato costat for moldg materal(mutes/ or secods/cm ), V rser volume ( 3 or cm 3 ), S coolg surface area of the rser. 109

111 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag he obectve s to desg the smallest rser such that.where soldfcato tme of the rser, soldfcato tme of the castg, tc / / K V S K V S (60) R R R C C C he rser ad castg are assumed to be molded the same materal so that KR ad K C are equal.so V / S V / S. (61) R R C C he castg has a specfed volume ad surface area, so VC / SC Y costat, whch s called the castg modulus. he, V / S Y, V C C R DR H R / 4, S D H D / 4 R R R R t R t C tr (6) herefore / 4 / 4 / 4 We take D D H D D H H D Y (63) V 96 C DR HR 1 (64) R R R R R R R R cubc ch. S square ch. C herefore Mult-obectve cyldrcal rser desg problem ca be stated as (P6), / 4 (65) Mmze V D H D H R R R R R, / 4 (66) Mmze t D H D H H D Subect to R R R R R R R DR HR 1 (67) D, H 0 (68) R R Here pay-off matrx s D H R R V R R able : Values of Optmal Decso Varables ad Obectve Fuctos by Neutrosophc Optmzato echque Optmal Decso varables D * R H * R Optmal Obectve Fuctos V D, H * * * R R R t D, H * * * R R R sprato levels of ruth, Falsty ad Idetermacy Membershp Fuctos * * *

112 Neutrosophc Operatoal Research Volume II 6 Coclusos ad Future Work I vew of comparg the eutrosophc optmzato wth tutstc fuzzy optmzato method, we also obtaed the soluto of the umercal problem by tutostc fuzzy optmzato method [14] ad took the best result obtaed for comparso wth preset study s to gve the effectve algorthm for eutrosophc optmzato method for gettg optmal solutos to a Mult-obectve o-lear programmg problem. he comparsos of results obtaed for udertake problem clearly show the superorty of Neutrosophc Optmzato over Itutostc Fuzzy Optmzato. Fally, as a applcato of Neutrosophc Mult-obectve Rser Desg Problem s preseted ad usg Neutrosophc Optmzato algorthm a optmzato algorthm s obtaed. ckowledgemet he authors would lke to thak the aoymous referees for ther costructve commets ad suggestos that have helped a lot to come up wth ths mproved form of the chapter. Refereces [1] Zadeh, L.Fuzzy Sets, Iform ad Cotrol1965:8: [] urkse, I. B. Iterval valued fuzzy sets based o ormal forms. Fuzzy sets ad systems.1986: 0(): [3] taassov, K.. Itutostc fuzzy sets. Fuzzy sets ad Systems.1986: 0(1): [4] Smaradache, F., Ufyg Feld Logcs Neutrosophy: Neutrosphc Probablty, Set ad Logc, Rehoboth, merca Research Press. [5] taassov, K.. More o tutostc fuzzy sets. Fuzzy sets ad systems. 1989:33(1): [6] Wag, H., Smaradache, F., Zhag, Y., & Suderrama, R. Sgle valued eutrosophc sets. Revew of the r Force cademy.010: (1): 10. [7] Smaradache, F. Neutrosophc set-a geeralzato of the tutostc fuzzy set. Iteratoal Joural of Pure ad ppled Mathematcs, 005:4(3): 87. [8] Bharat, S. K., & Sgh, S. R. Solvg mult obectve lear programmg problems usg tutostc fuzzy optmzato method: a comparatve study. Iteratoal Joural of Modelg ad Optmzato.014: 4(1):10. [9] Jaa, B., & Roy,. K. Mult-obectve tutostc fuzzy lear programmg ad ts applcato trasportato model. Notes o Itutostc Fuzzy Sets.007: 13(1): [10] Mahapatra, G. S., Mtra, M., & Roy,. K. Itutostc fuzzy mult-obectve mathematcal programmg o relablty optmzato model. Iteratoal oural of fuzzy systems.010: 1(3): [11] Zmmerma, H. J. Fuzzy programmg ad lear programmg wth several obectve fuctos. Fuzzy sets ad systems1978: 1(1): [1] Chvorov, N. Cotrol of the Soldfcato of Castgs by Calculato. Foudry rade Joural.1939:70(8): [13] Creese, R. C. Optmal rser desg by geometrc programmg. FS cast metals research oural.1971: 7():

113 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [14] Creese, R. C. Dmesog of rser for log freezg rage alloys by geometrc programmg. FS cast metals research oural1971: 7(4): [15] gelov, P. P. Optmzato a tutostc fuzzy evromet. Fuzzy sets ad Systems.1997:86(3): [16] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [17] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [18] bdel-basset, Mohamed, Ma Mohamed, ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [19] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [0] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [1] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [3] bdel-baset, Mohamed, Ibrahm M. Hezam, ad Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [4] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [5] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016): [6] Mrdula Sarkar, Samr Dey, apa Kumar Roy: russ Desg Optmzato usg Neutrosophc Optmzato echque, Neutrosophc Sets ad Systems, vol. 13, 016, pp do.org/10.581/zeodo [7] Ptu Das, apa Kumar Roy: Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem, Neutrosophc Sets ad Systems, vol. 9, 015, pp do.org/10.581/zeodo [8] Naga Rau I, Raeswara Reddy P, Dr. Dwakar Reddy V, Dr. Krshaah G: Real Lfe Decso Optmzato Model, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [9] Wezhog Jag, Ju Ye: Optmal Desg of russ Structures Usg a Neutrosophc Number Optmzato Model uder a Idetermate Evromet, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [30] Moa Gamal Gafar, Ibrahm El-Heawy: Itegrated Framework of Optmzato echque ad Iformato heory Measures for Modelg Neutrosophc Varables, Neutrosophc Sets ad Systems, vol. 15, 017, pp do.org/10.581/zeodo

114 Neutrosophc Operatoal Research Volume II VI russ Desg Optmzato usg Neutrosophc Optmzato echque Mrdula Sarkar 1 apa Kumar Roy 1Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah , Ida. Emal: mrdula.sarkar86@redffmal.com Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah , Ida. Emal: roy_t_k@yahoo.co. bstract I ths paper, we develop a eutrosophc optmzato (NSO) approach for optmzg the desg of plae truss structure wth sgle obectve subect to a specfed set of costrats. I ths optmum desg formulato, the obectve fuctos are the weght of the truss ad the deflecto of loaded ot; the desg varables are the cross-sectos of the truss members; the costrats are the stresses members. classcal truss optmzato example s preseted here to demostrate the effcecy of the eutrosophc optmzato approach. he test problem cludes a two-bar plaar truss subected to a sgle load codto. hs sgle-obectve structural optmzato model s solved by fuzzy ad tutostc fuzzy optmzato approach as well as eutrosophc optmzato approach. Numercal example s gve to llustrate our NSO approach. he result shows that the NSO approach s very effcet fdg the best dscovered optmal solutos. Keywords Neutrosophc Set; Sgle Valued Neutrosophc Set; Neutrosophc Optmzato; No-lear Membershp Fucto; Structural Optmzato. 1 Itroducto I the feld of cvl egeerg, olear structural desg optmzatos are of great of mportace. herefore, the descrpto of structural geometry ad mechacal propertes lke stffess are requred for a structural system. However, 113

115 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag the system descrpto ad system puts may ot be exact due to huma errors or some uexpected stuatos. t ths ucture fuzzy set theory provdes a method whch deal wth ambguous stuatos lke vague parameters, o-exact obectve ad costrat. I structural egeerg desg problems, the put data ad parameters are ofte fuzzy/mprecse wth olear characterstcs that ecesstate the developmet of fuzzy optmum structural desg method. Fuzzy set (FS) theory has log bee troduced to hadle exact ad mprecse data by Zadeh [], Later o Bellma ad Zadeh [4] used the fuzzy set theory to the decso makg problem. he fuzzy set theory also foud applcato structural desg. Several researchers lke Wag et al. [8] frst appled α-cut method to structural desgs where the o-lear problems were solved wth varous desg levels α, ad the a sequece of solutos were obtaed by settg dfferet level-cut value of α. Rao [3] appled the same α-cut method to desg a four bar mechasm for fucto geeratg problem. Structural optmzato wth fuzzy parameters was developed by Yeh et al. [9]. Xu [10] used two-phase method for fuzzy optmzato of structures. Shh et al. [5] used level-cut approach of the frst ad secod kd for structural desg optmzato problems wth fuzzy resources. Shh et al. [6] developed a alteratve α-level-cuts methods for optmum structural desg wth fuzzy resources. Dey et al. [11] used geeralzed fuzzy umber cotext of a structural desg. Dey et al used basc t-orm based fuzzy optmzato techque for optmzato of structure. Dey et al. [13] developed parameterzed t-orm based fuzzy optmzato method for optmum structural desg. lso, Dey et.al [14] Optmzed shape desg of structural model wth mprecse coeffcet by parametrc geometrc programmg. I such exteso, taassov [1] troduced Itutostc fuzzy set (IFS) whch s oe of the geeralzatos of fuzzy set theory ad s characterzed by a membershp fucto, a o- membershp fucto ad a hestacy fucto. I fuzzy sets the degree of acceptace s oly cosdered but IFS s characterzed by a membershp fucto ad a o-membershp fucto so that the sum of both values s less tha oe. trasportato model was solved by Jaa et al.[15]usg mult-obectve tutostc fuzzy lear programmg. Dey et al. [1] solved two bar truss o-lear problem by usg tutostc fuzzy optmzato problem. Dey et al. [16] used tutostc fuzzy optmzato techque for mult obectve optmum structural desg. Itutostc fuzzy sets cosder both truth membershp ad falsty membershp. Itutostc fuzzy sets ca oly hadle complete formato ot the determate formato ad cosstet formato.i eutrosophc sets determacy s quatfed explctly ad truth membershp, determacy membershp ad falsty membershp are depedet. Neutrosophc theory was troduced by Smaradache [7,18-7]. he motvato of the preset study s to gve computatoal algorthm for solvg mult-obectve structural problem by sgle 114

116 Neutrosophc Operatoal Research Volume II valued eutrosophc optmzato approach. Neutrosophc optmzato techque s very rare applcato to structural optmzato. We also am to study the mpact of truth membershp, determacy membershp ad falsty membershp fucto such optmzato process. he results are compared umercally both fuzzy optmzato techque, tutostc fuzzy optmzato techque ad eutrosophc optmzato techque. From our umercal result, t s clear that eutrosophc optmzato techque provdes better results tha fuzzy optmzato ad tutostc fuzzy optmzato. Sgle-obectve Structural Model I szg optmzato problems, the am s to mmze sgle obectve fucto, usually the weght of the structure uder certa behavoral costrats o costrat ad dsplacemet. he desg varables are most frequetly chose to be dmesos of the cross sectoal areas of the members of the structures. Due to fabrcatos lmtatos the desg varables are ot cotuous but dscrete for beloggess of cross-sectos to a certa set. dscrete structural optmzato problem ca be formulated the followg form Mmze W (1) 0, 1,,..., () subect to m d R, 1,,..., (3) where W represets obectve fucto, s the behavoral costrats, m ad are the umber of costrats ad desg varables respectvely. gve R set of dscrete value s expressed by ad ths paper obectve fucto s m take asw 1 l ad costrat are chose to be stress of structures as follows wth allowable tolerace for 1,,..., mwhere ad l are weght of ut volume ad legth of umber of structural elemet, respectvely. Mathematcal Prelmares ad d 0 0 th elemet respectvely, are the m s the th stress,allowable stress I the followg, we brefly descrbe some basc cocepts ad basc operatoal laws related to eutrosophc set.1 Fuzzy Set (FS) Let X be a fxed set. fuzzy set of X s a obect havg the form 115

117 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag (4) x x x X, where the fucto : 0,1 x X to the set... Itutostc Fuzzy Set (IFS) X stads for the truth membershp of the elemet Let a set X be fxed. tutostc fuzzy set or IFS X s a obect of the form X, x, F x x X (5) where : X 0,1 ad : 0,1 F X defe the truth membershp ad falsty membershp respectvely, for every elemet of 0 x F x Sgle-Valued Neutrosophc Set (SVNS) x X such that Let a set X be the uverse of dscourse. sgle valued eutrosophc set over X s a obect havg the form,,, (6) x x I x F x x X : X 0,1, : 0,1 where I X ad : 0,1 F X are truth, determacy ad falsty membershp fuctos respectvely so as to 0 x I x F x 3 for all x X..4. Uo of Neutrosophc Sets (NSs) he uo of two sgle valued eutrosophc sets valued eutrosophc set deoted by U ad B s a sgle U B x, x, I x, F x x X (7) U U U ad s defed by the followg codtos () x x x max,, U B () I x I x I x max,, U B () F x m F x, F x for all x X U B for ype-i Or aother way by defg followg codtos 116

118 Neutrosophc Operatoal Research Volume II () x x x max,, U B () I x m I x, I x U B () F x m F x, F x for all x X U B for ype-ii where U, x I U x, F U x represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be calculated for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.6, 0.4, 0. / x 0.5, 0., 0.3 / x 0.7, 0., 0. / x (8) ad for ype -II as 1 3 B 0.6, 0.1, 0. / x 0.5, 0., 0.3 / x 0.7, 0.1, 0. / x (9).5. Itersecto of Neutrosophc Sets 1 3 he tersecto of two sgle valued eutrosophc sets sgle valued eutrosophc set E s deoted by ad,,, (10) E B x x I x F x x X E E E B s a ad s defed by the followg codtos () x x x m,, E B () I x I x I x m,, E B () F x max F x, F x for all x X E B for ype-i Or aother way by defg followg codtos () x x x m,, E B () I x max I x, I x E B 117

119 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag () F x max F x, F x for all x X E B for ype-ii where E x, I E x, F E x represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be measured for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.3, 0.1, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x (11) ad for ype -II as 1 3 B 0.3, 0.4, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0., 0.5 / x (1) Mathematcal alyss 3.1. Neutrosophc Optmzato echque to Solve Mmzato ype Sgle-Obectve No-lear Programmg Problem Let a olear sgle-obectve optmzato problem be Mmze f x (13) Such that 1,,..., g x b m x 0 Usually costrats goals are cosdered as fxed quatty.but real lfe problem, the costrat goal caot be always exact. herefore, we ca cosder the costrat goal for less tha type costrats at least ad t may possble to exted to 0 b b.hs fact seems to take the costrat goal as a eutrosophc fuzzy set ad whch wll be more realstc descrptos tha others. he the NLP becomes NSO problem wth eutrosophc resources, whch ca be descrbed as follows Mmze f x (14) Such that b 118

120 Neutrosophc Operatoal Research Volume II 1,,..., g x b m x 0 o solve the NSO (13), we are presetg a soluto procedure for sgleobectve NSO problem (13) as follows Step-1: Followg warer s approach solve the sgle obectve o-lear programmg problem wthout tolerace costrats e g x b., wth 0 tolerace of acceptace costrats (.e ) by approprate olear programmg techque Here they are Sub-problem-1 g x b b Mmze f x (15) Such that 1,,..., g x b m x 0 Sub-problem- Mmze f x (16) Such that 0, 1,,..., g x b b m x 0 We may get optmal soluto x * x 1, f x * f x 1 x x, f x f x * 1 * 1 ad Step-: From the result of step 1 we ow fd the lower boud ad upper boud of obectve fuctos. IfU, U I, U F f x f x f x be the upper bouds of truth, I F determacy, falsty fucto for the obectve respectvely ad L, L, L f x f x f x be the lower boud of truth, determacy, falsty membershp fuctos of obectve respectvely. the 119

121 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag U U, L L where 0 U L F F f x f x f x f x f x f x f x f x U U, L L where 0 U L F F f x f x f x f x f x f x f x f x L L, U L where 0 U L I I f x f x f x f x f x f x f x f x Step-3: I ths step we calculate membershp for truth, determacy ad falsty membershp fucto of obectve as follows 1 f f x Lf x U f x f x f x 1 exp f x f L f x U f x f x U f x L f x 0 f f x U f x I 1 f f x Lf x I U f x f x I I I f x exp f x I I f L f x U f x f x U f x L f x I 0 f f x U f x F 0 f f x Lf x 1 1 F F U L f x f x tah F F F f x f x f x f x f Lf x f x U f x F 1 f f x U f x (17) (18) (19) where, are o-zero parameters prescrbed by the decso maker. Step-4: I ths step usg expoetal ad hyperbolc membershp fucto we calculate truth, determacy ad falsty membershp fucto for costrats as follows 1 f g x b U g x g x 0 g 1 exp g x x f b g x b b Ug x L g x 0 0 f g x b b 1 f g x b b g x g x I g exp g x x f b g x b g x g x 0 f g x b g x (0) (1) 10

122 Neutrosophc Operatoal Research Volume II 0 f g x b g x b b g 0 tah x F g g x x g x f b g x g g x x b b 0 1 f g x b b () where, are o-zero parameters prescrbed by the decso maker ad for 1,,..., m 0, b. 0 g x g x Step-5: Now usg NSO for sgle obectve optmzato techque the optmzato problem (13) ca be formulated as Maxmze Such that ; f x (3) x (4) x ; g ; f x (5) I x (6) g ; I x (7) F x (8) ; f x g ; F x (9) 3; (30) ; ; (31) where,, 0,1 (3) x m f x, g x (33) D for 1,,..., m f x g x I x m I f x, I g x (34) D for 1,,..., m ad f x g x F x m F f x, F g x for 1,,..., m (35) D f x g x 11

123 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag are the truth,determacy ad falsty membershp fucto of decso set D f x g x m 1. Now f o-lear membershp be cosdered the above problem (3-35) ca be reduced to followg crsp lear programmg problem Maxmze Such that (36) U L f x f x f x U ; (37) f x U ; f x f x (38) f x f x f x U L f x f x f x ; (39) 0 b g x b b 0 ; (40) 0 g x g x g x b ; (41) g x gx 0 b b g x ; (4) 3; (43) ; ; (44),, 0,1 (45) where l 1 ; (46) 4; (47) f x g x U l ; b F f x L F f x ;, for 1,,..., m (48) (49) (50) 1 tah 1. (51) 1

124 Neutrosophc Operatoal Research Volume II hs crsp olear programmg problem ca be solved by approprate mathematcal algorthm. 4 Soluto of Sgle-obectve Structural Optmzato Problem (SOSOP) by Neutrosophc Optmzato echque o solve the SOSOP (1), step 1 of 3 s used ad we wll get optmum solutos of two sub problem as ad.fter that accordg to step we fd upper ad lower boud of membershp fucto of obectve fucto as I F UW, UW, UW ad U, U I, U F W W W where W U max W, W, L m W, W, (5) W U U, L L where 0 U L (53) F F W W W W W W W W L L, U L where 0 U L (54) I I W W W W W W W W Let the o-lear membershp fucto for obectve fucto W be W 1 f W L U W W 1 exp f L W U UW L W 0 f W U W W W W W (55) I W 1 f W L I U W I W exp I I f L W U UW L W I 0 f W UW W I I W W W (56) 13

125 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag g x exp x 1 f x I f 0 f x 0 f b b 0 tah F f 0 1 f (57) (58) where are, o-zero parameters prescrbed by the decso maker ad for m 0 1,,..., 0,, the eutrosophc optmzato problem ca be formulated as such that Max W (59) ; W (60) ; (61) W ; I W (6) ; I (63) W ; F W (64) F (65) 3;, ; (66),, 0,1 (67) he above problem ca be reduced to followg crsp lear programmg problem, for o-lear membershp as Maxmze such that (68) U L W W W U ; (69) W 14

126 Neutrosophc Operatoal Research Volume II U L W W W W ; (70) W W U ; (71) W W 0 0 ; (7) 0 ; (73) 0 ; (74) 3; ; ; (75),, 0,1 (76) where l 1 ; (77) 4; (78) W l ; U F W 6 L F W ; (79) (80) 1 tah 1. (81) ad U F 6 L F ; (8) hs crsp olear programmg problem ca be solved by approprate mathematcal algorthm. 5 Numercal llustrato well-kow two-bar [17] plaar truss structure s cosdered. he desg obectve s to mmze weght of the structural W 1,, yb of a statstcally loaded two-bar plaar truss subected to stress 1,, ybcostrats o each of the truss members 1,. 15

127 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Fgure 1. Desg of the two-bar plaar truss where he mult-obectve optmzato problem ca be stated as follows Mmze W 1,, yb 1 xb l yb xb y (83) B Such that B 1 P x l y,, y ; (84) BC 1 P B B B B l1 P x y,, y ; (85) B B B C BC l 0.5 y B 1.5 (86) 0, 0; 1 odal load ; perpedcular dstace from C to pot B volume desty ; l legth of C ; x. 1 Cross secto of bar- B ; Cross secto of bar- BC. maxmum allowable tesle stress, C maxmum allowable compressve stress ad y B y-co-ordate of ode B. Iput data for crsp model (10) s table 1. Soluto : ccordg to step of 4,we fd upper ad lower boud of membershp fucto of obectve fucto as B 16

128 Neutrosophc Operatoal Research Volume II U, U, U I F W W W ad L, L I, L F W W W where U L F UW (87) W I LW (88) W L F W ,, W W (89) I U L 0 W W W W (90) Now usg the bouds we calculate the membershp fuctos for obectve as follows 1 f W 1,, yb W 1,, y B W 1,, 1,, y 1 exp ,, W yb B f W yb f W 1,, yb f W 1,, yb W,, y I W,, y exp f W,, y W 1,, yb 0 f W 1,, yb W W 1 B 1 B 1 B W W (91) (9) 0 f W 1,, yb W W F W 1,, 1,, y tah 1,, ,, W yb B W yb f W W yb W 1 f W 1,, yb (93) Smlarly the membershp fuctos for tesle stress are 1 f 1,, yb ,, y B 1,, 1,, 1 exp ,, 150 yb yb f y B 0 0 f 1,, yb f 1,, yb , y B I 1,, 1,, exp 130 1,, 130 yb yb f y B 0 f 1,, yb 130 (94) (95) 17

129 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag 0 f 1,, yb F 1,, 1,, tah 1,, 130 1,, 150 yb yb yb f y B 0 1 f 1,, yb 150 where 0, 0 (96) ad the membershp fuctos for compressve stress costrat are 1 f C 1,, yb C 1,, y B 1,, 1,, 1 exp ,, 100 C yb C yb f C y B 10 0 f C 1,, yb f C 1,, yb ,, C C y B I 1,, 1,, exp 90 1,, 90 C yb C yb f C y B C C 0 f C 1,, yb 90 C 0 f C 1,, yb 90 C C F 1,, 1,, tah 1,, 90 1,, 100 C yb C yb C yb f C C y B 10 C 1 f C 1,, yb 100 (97) (98) (99) where 0, 10. C C Now, usg above metoed truth, determacy ad falsty membershp fucto NLP (83-86) ca be solved by NSO techque for dfferet values of ad,,. he optmum soluto of SOSOP(83-86) s gve,, W C W C table () ad the soluto s compared wth fuzzy ad tutostc fuzzy optmzato techque able 1: Iput data for crsp model (83-86) ppled load P KN Volume desty 3 KN / m Legth l m Maxmum allowable tesle stress Mpa wth tolerace 0 Maxmum allowable compressve stress Mpa C 90 wth tolerace 10 Dstace of x B from C m 1 18

130 Neutrosophc Operatoal Research Volume II able : Comparso of Optmal soluto of SOSOP (83-86) based o dfferet methods Methods Fuzzy sgle-obectve olear programmg (FSONLP) wth o-lear membershp fuctos Itutostc fuzzy sgleobectve olear programmg (IFSONLP) wth o-lear membershp fuctos 0.8, 16, Neutosophc optmzato(nso) wth o-lear membershp fuctos 0.8, 16, , 8, m m, KN W yb m Here we get best solutos for the dfferet tolerace determacy expoetal membershp fucto of obectve fuctos for ths structural optmzato problem. From the table, t shows that NSO techque gves better Pareto optmal result the perspectve of Structural Optmzato. 7 Coclusos he ma obectve of ths work s to llustrate how eutrosophc optmzato techque usg o-lear membershp fucto ca be utlzed to solve a olear structural problem. he cocept of eutrosophc optmzato techque allows oe to defe a degree of truth membershp, whch s ot a complemet of degree of falsty; rather, they are depedet wth degree of determacy. he umercal llustrato shows the superorty of eutrosophc optmzato over fuzzy optmzato as well as tutostc fuzzy optmzato. he results of ths study may lead to the developmet of effectve eutrosophc techque for solvg other model of olear programmg problem other egeerg feld. 1, ad 3 for 19

131 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag ckowledgemet he authors would lke to thak the aoymous referees for ther costructve commets ad suggestos that have helped a lot to come up wth ths mproved form of the chapter. Refereces [1] taassov, K.. Itutostc fuzzy sets. Fuzzy Sets ad Systems.1986:0(1): [] Zadeh, L.. Fuzzy set. Iformato ad Cotrol. 1965):8(3): [3] Rao, S. S. Descrpto ad optmum desg of fuzzy mechacal systems. Joural of Mechasms, rasmssos, ad utomato Desg.1987:109(1) [4] Bellma, R. E., &Zadeh, L.. Decso-makg a fuzzy evromet. Maagemet scece.1970:17(4):b-141. [5] Shh, C. J., & Lee, H. W. Level-cut approaches of frst ad secod kd for uque soluto desg fuzzy egeerg optmzato problems. amkag Joural of Scece ad Egeerg.004:7(3): [6] Shh, C. J., Ch, C. C., & Hsao, J. H. lteratve α-level-cuts methods for optmum structural desg wth fuzzy resources. Computers & structures.003:81(8): [7] Smaradache, F. Neutrosophy, eutrosophc probablty, set ad logc, mer. Res. Press, Rehoboth, US. 1995:105. [8] Wag, G.Y. & Wag, W.Q. Fuzzy optmum desg of structure. Egeerg Optmzato.1985: [9] Yeh, Y.C. & Hsu, D.S. Structural optmzato wth fuzzy parameters. Computer ad Structure. 1990:37(6): [10] Chagwe, X. Fuzzy optmzato of structures by the two-phase method. Computers & Structures. 1989:31(4): [11] Dey, S., & Roy,. K. Fuzzy programmg echque for Solvg Multobectve Structural Problem. Iteratoal Joural of Egeerg ad Maufacturg, 014:4(5):4. [1] Dey, S., & Roy,. K. Optmzed soluto of two bar truss desg usg tutostc fuzzy optmzato techque. Iteratoal Joural of Iformato Egeerg ad Electroc Busess.014:6(4):45. [13] Dey, S., & Roy,. K. Mult-obectve structural desg problem optmzato usg parameterzed t-orm based fuzzy optmzato programmg echque. Joural of Itellget ad Fuzzy Systems.016:30(): [14] Dey, S., & Roy,. Optmum shape desg of structural model wth mprecse coeffcet by parametrc geometrc programmg. Decso Scece Letters.015:4(3): [15] Jaa, B., & Roy,. K. Mult-obectve tutostc fuzzy lear programmg ad ts applcato trasportato model. Notes o Itutostc Fuzzy Sets.007:13(1): [16] Dey, S.,& Roy,. K. Mult-obectve structural optmzato usg fuzzy ad tutostc fuzzy optmzato techque. Iteratoal Joural of Itellget systems ad applcatos.015:7(5):57. [17] l, N., Behda, K., & Fawaz, Z. pplcablty ad vablty of a G based fte elemet aalyss archtecture for structural desg optmzato. Computers & structures.003:81():

132 Neutrosophc Operatoal Research Volume II [18] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [19] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [0] bdel-basset, Mohamed, Ma Mohamed, ad ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [1] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [3] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [4] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [5] bdel-baset, Mohamed, Ibrahm M. Hezam, ad Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [6] Mrdula Sarkar, Samr Dey, apa Kumar Roy: russ Desg Optmzato usg Neutrosophc Optmzato echque, Neutrosophc Sets ad Systems, vol. 13, 016, pp do.org/10.581/zeodo [7] Ptu Das, apa Kumar Roy: Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem, Neutrosophc Sets ad Systems, vol. 9, 015, pp do.org/10.581/zeodo [8] Naga Rau I, Raeswara Reddy P, Dr. Dwakar Reddy V, Dr. Krshaah G: Real Lfe Decso Optmzato Model, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [9] Wezhog Jag, Ju Ye: Optmal Desg of russ Structures Usg a Neutrosophc Number Optmzato Model uder a Idetermate Evromet, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [30] Moa Gamal Gafar, Ibrahm El-Heawy: Itegrated Framework of Optmzato echque ad Iformato heory Measures for Modelg Neutrosophc Varables, Neutrosophc Sets ad Systems, vol. 15, 017, pp do.org/10.581/zeodo [31] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [3] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016):

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134 Neutrosophc Operatoal Research Volume II VII Mult-obectve Neutrosophc Optmzato echque ad ts pplcato to Structural Desg Mrdula Sarkar 1 apa Kumar Roy 1Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah , Ida. Emal: mrdula.sarkar86@redffmal.com Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah , Ida. Emal: roy_t_k@yahoo.co. bstract I ths paper, we develop a mult-obectve o-lear eutrosophc optmzato (NSO) approach for optmzg the desg of plae truss structure wth multple obectves subect to a specfed set of costrats. I ths optmum desg formulato, the obectve fuctos are the weght of the truss ad the deflecto of loaded ot; the desg varables are the cross-sectos of the truss members; the costrats are the stresses members. classcal truss optmzato example s preseted here to demostrate the effcecy of the eutrosophc optmzato approach. he test problem cludes a three-bar plaar truss subected to a sgle load codto.hs mult-obectve structural optmzato model s solved by eutrosophc optmzato approach. Wth lear ad olear membershp fucto. Numercal example s gve to llustrate our NSO approach. Keywords Neutrosophc Set, Sgle Valued Neutrosophc Set, Neutrosophc Optmzato, Structural model. 1 Itroducto he research area of optmal structural desg has bee recevg creasg atteto fromboth academa ad dustry over the past three decades order to mprove structural performace ad to reduce desg costs. However, the real world, ucertaty or vagueess s prevalet the Egeerg 133

135 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Computatos. I the cotext of structural desg the ucertaty s coected wth lack of accurate data of desg factors. hs tedecy has bee chagg due to the crease the use of fuzzy mathematcal algorthm for dealg wth ths class of problems.fuzzy set (FS) theory has log bee troduced to hadle exact ad mprecse data by Zadeh [], Later o Bellma ad Zadeh [4] used the fuzzy set theory to the decso makg problem. he fuzzy set theory also foud applcato structural desg. Several researchers lke Wag et al. [8] frst appled α-cut method to structural desgs where the o-lear problems were solved wth varous desg levels α, ad the a sequece of solutos were obtaed by settg dfferet level-cut value of α. Rao [3] appled the same α-cut method to desg a four bar mechasm for fucto geeratg problem. Structural optmzato wth fuzzy parameters was developed by Yeh et al. [9]. Xu [10] used two-phase method for fuzzy optmzato of structures. Shh et al. [5] used level-cut approach of the frst ad secod kd for structural desg optmzato problems wth fuzzy resources. Shh et al. [6] developed a alteratve α-level-cuts methods for optmum structural desg wth fuzzy resources. Dey et al. [11] used geeralzed fuzzy umber cotext of a structural desg. Dey et al.[14]used basc t-orm based fuzzy optmzato techque for optmzato of structure. Dey et al. [13] developed parameterzed t-orm based fuzzy optmzato method for optmum structural desg. I such exteso, taassov [1] troduced Itutostc fuzzy set (IFS) whch s oe of the geeralzatos of fuzzy set theory ad s characterzed by a membershp fucto, a o- membershp fucto ad a hestacy fucto. I fuzzy sets the degree of acceptace s oly cosdered but IFS s characterzed by a membershp fucto ad a o-membershp fucto so that the sum of both values s less tha oe. trasportato model was solved by Jaa et al.[15]usg multobectve tutostc fuzzy lear programmg. Dey et al. [1] solved two bar truss o-lear problem by usg tutostc fuzzy optmzato problem. Dey et al. [16] used tutostc fuzzy optmzato techque for mult obectve optmum structural desg. Itutostc fuzzy sets cosder both truth membershp ad falsty membershp. Itutostc fuzzy sets ca oly hadle complete formato ot the determate formato ad cosstet formato.i eutrosophc sets determacy s quatfed explctly ad truth membershp, determacy membershp ad falsty membershp whch are depedet. Neutrosophc theory was troduced by Smaradache [7,17-6]. he motvato of the preset study s to gve computatoal algorthm for solvg mult-obectve structural problem by sgle valued eutrosophc optmzato approach. Neutrosophc optmzato techque s very rare applcato to structural optmzato. We also am to study the mpact of truth expoetal membershp, determacy expoetal membershp ad falsty hyperbolc membershp fucto such optmzato process. he results are compared 134

136 Neutrosophc Operatoal Research Volume II umercally lear ad olear eutrosophc optmzato techque. From our umercal result, t has bee see that there s o chage betwee the result of lear ad o-lear eutrosophc optmzato techque the perspectve of structural optmzato techque. Mult-obectve Structural Model I the desg problem of the structure.e. lghtest weght of the structure ad mmum deflecto of the loaded ot that satsfes all stress costrats members of the structure. I truss structure system,the basc parameters (cludg allowable stress,etc) are kow ad the optmzato s target s that detfy the optmal bar truss cross-secto area so that the structure s of the smallest total weght wth mmum odes dsplacemet a gve load codtos. he mult-obectve structural model ca be expressed as Mmze W (1) mmze subect to m max () (3) 1 where,,... are the desg varables for the cross secto, s the group umber of desg varables for the cross secto bar, W L s the total weght of the structure, s the deflecto of the loaded ot,where ad are the bar legth, cross secto area ad desty of the group bars respectvely. s the stress costrat ad s allowable stress of the group L, bars uder varous codtos, cross secto area respectvely m 3 Mathematcal Prelmares ad th 1 max are the lower ad upper bouds of I the followg, we brefly descrbe some basc cocepts ad basc operatoal laws related to eutrosophc set.1 Fuzzy Set(FS) Let X be a fxed set. fuzzy set of X s a obect havg the form (4) x x x X, 135

137 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag where the fucto : 0,1 elemet x X to the set... Itutostc Fuzzy Set(IFS) X stads for the truth membershp of the Let a set X be fxed. tutostc fuzzy set or IFS X s a obect of the form X, x, F x x X (5) where : X 0,1 ad : 0,1 F X defe the truth membershp ad falsty membershp respectvely, for every elemet of x X such that 0 x F x Sgle-Valued Neutrosophc Set(SVNS) Let a set be the uverse of dscourse. sgle valued eutrosophc set over X s a obect havg the form X,,, (6) x x I x F x x X where : 0,1, : 0,1 F : X 0,1 are truth, determacy ad falsty membershp fuctos respectvely so as to 0 x I x F x 3 for all x X. X I X ad.4. Uo of Neutrosophc Sets(NSs) he uo of two sgle valued eutrosophc sets valued eutrosophc set deoted by U ad B s a sgle U B x, x, I x, F x x X (7) U U U ad s defed by the followg codtos () x x x max,, U B () I x I x I x max,, U B () F x m F x, F x for all x X U B Or aother way by defg followg codtos () x x x max,, U B () I x m I x, I x U B for ype-i 136

138 Neutrosophc Operatoal Research Volume II where () F x m F x, F x for all x X U U B, x I U x, F U x for ype-ii represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example : Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be calculated for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.6, 0.4, 0. / x 0.5, 0., 0.3 / x 0.7, 0., 0. / x (8) ad for ype -II as 1 3 B 0.6, 0.1, 0. / x 0.5, 0., 0.3 / x 0.7, 0.1, 0. / x (9) Itersecto of Neutrosophc Sets he tersecto of two sgle valued eutrosophc sets a sgle valued eutrosophc set E s deoted by ad,,, (10) E B x x I x F x x X E E E ad s defed by the followg codtos () x x x m,, E B () I x I x I x m,, E B () F x max F x, F x for all x X E B for ype-i Or aother way by defg followg codtos () x x x m,, E B () I x max I x, I x E B () F x max F x, F x for all x X E B for ype-ii where, I, x F E E E x represet truth membershp, determacy-membershp ad falsty-membershp fuctos of uo of eutrosophc sets B s 137

139 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be measured for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.3, 0.1, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x (11) ad for ype -II as 1 3 B 0.3, 0.4, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0., 0.5 / x (1) Mathematcal alyss 4.1. Neutrosophc Optmzato echque to Solve Mmzato ype Mult-Obectve No-lear Programmg Problem Decso makg s a process of solvg the problem volvg the goals uder costrats.he outcome s a decso whch should a acto.decso makg plays a mportat role egeerg scece.it s dffcult process due to factors lke complete ad mprecse formato whch ted to preseted real lfe stuatos.i the decso makg process,our ma target s to fd the value from the selected set wth the hghest degree of membershp the decso set ad these values support the goals uder costrats oly.but there may be stuatos where some values from selected set caot support.e such values strogly agast the goals uder costrats whch are o-admssble.i ths case we fd such values from selected set wth last degree of o membershp the decso sets.itutostc fuzzy sets ca oly hadle complete formato ot the determate formato ad cosstet formato whch exsts commoly belef systems.i eutrosophc set,determacy s quatfed explctely ad truth membershp,determacy membershp ad falsty membershp are depedet.so t s atural to adopt the purpose the value from the selected set wth hghest degree of truth-membershp,determacymembershp ad least degree of falsty membershp o the decso set.hese factors dcate that a decso makg process takes place eutrosophc evromet. olear mult-obectve optmzato of the problem s of the form Mmze f1 x, f x,..., f p x (13) 1,,..., (14) g x b q Now the decso set costrats s defed D, a coucto of Neutrosophc obectves ad 138

140 Neutrosophc Operatoal Research Volume II p q D Gk C x x I x F x D D D k1 1 Here,, x, x, x,..., x ; G1 G G3 G p x m D for all x X x, x, x,..., x C1 C C3 C q I x, I x, I x,..., I x ; G1 G G3 G p I x m D for all x X I x, I x, I x,..., I x C1 C C3 C q F x, F x, F x,..., F x ; G1 G G3 G p F x m D for all x X F x, F x, F x,..., F x C1 C C3 C q Where,, D D D (15) (16) (17) (18) x I x F x are truth-membershp fucto, determacy membershp fucto,falsty membershp fucto of eutrosophc decso set respectvely.now usg the eutrosophc optmzato, problem (13-14) s trasformed to the o-lear programmg problem as Max (19) Max (0) M (1) such that G k ; x () C ; x (3) G k ; I x (4) C ; I x (5) G k ; F x (6) C ; F x (7) 3; (8) ; ; (9),, 0,1 (30) 139

141 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Now ths o-lear programmg problem (19-30) ca be easly solved by a approprate mathematcal programmg to gve soluto of mult-obectve o-lear programmg problem (13-14) by eutrosophc optmzato approach. Computatoal lgorthm Step-1: Solve the MONLP problem (13-14) as a sgle obectve olear problem p tmes for each problem by takg oe of the obectves at a tme ad gorg the others.hese soluto are kow as deal solutos. Let be the respectve optmal soluto for the dfferet obectve ad evaluate each obectve values for all these optmal soluto. k th k th Step-: From the result of step-1, determe the correspodg values for every obectve for each derved soluto, pay-off matrx ca be formulated as follows boud Step-3: For each obectve U k r* Uk max fk x ad k... p *... p f x f x f x f1 x f x f x f1 x f x f x * p p * p... p fk x fd lower boud L k x k ad the upper (31) L f x where r k (3) r* m k 1,,..., For truth membershp of obectves. Step-4:We represet upper ad lower bouds for determacy ad falsty membershp of obectves as follows : for k 1,,... p F F Uk Uk ad Lk Lk t U k Lk ; (33) I I k k k k k k L L ad U L s U L (34) Here tsare, predetermed real umbers Step-5: Defe truth membershp, determacy membershp ad falsty membershp fuctos as follows for k 1,,..., p 0,1 140

142 Neutrosophc Operatoal Research Volume II 1 f fk x Lfk x U f x f x f L f x U 0 f f x U fk x fk x k 1 exp k k f x fk x k fk x U fkx L fkx I 1 f fk x Lf x I U f k k x I fk x exp I I f L f x U U fkx L fkx I 0 f fk x U fk x f x I I fk x fk x fk x k (35) (36) F 0 f f x Lf x 1 1 F F U L f x f x tah F F F f x f x f x f L f x U f x f x f x F 1 f f x U f x (37) Step-6:Now eutosophc optmzato method for MONLP problem gves a equvalet olear programmg problem as: Maxmze such that k k ; (38) f x (39) k k ; I f x (40) k k ; F f x (41) 3; (4) ; ; (43),, 0,1 ; (44) 0, (45) g x b x k 1,,..., p; 1,,..., q whch s reduced to equvalet o lear programmg problem as Maxmze such that (46) 141

143 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag U L fk x L 4 f k k k ; (47) k k fk x ; k U L (48) fk ; 1,,..., f x L for k p (49) k fk k fk where log, log 1, (50) (51) 1 tah 1, (5) 4, (53) 6 L fk F F Uk k (54) 3; (55) ; ; (56),, 0,1 ; (57) ; g x b x 0, (58) hs crsp olear programmg problem ca be solved by approprate mathematcal algorthm. 5 Soluto of Mult-obectve Structural Optmzato Problem (MOSOP) by Neutrosophc Optmzato echque o solve the MOSOP (1), step 1 of 4s used.fter that accordg to step to pay off matrx s formulated. W * W W * I I ccordg to step- the boud of weght obectve UW, L W ; UW, L ad W F F UW, L for truth, determacy ad falsty membershp fucto respectvely. W he I I F F LW W UW ; LW W UW ; LW W U. Smlarly the W I I F F boud of deflecto obectve are U, L ; U, L ad U, L are respectvely for 14

144 Neutrosophc Operatoal Research Volume II truth, determacy ad falsty membershp fucto. he L U ; I I F F F L U; L U.Where L ; W LW W I U L (59) L I W L W ad L I, U F L, such that 0 W W W U U, L U F W U L ; (60) F L (61) I W UW LW ad 0 U L. herefore the truth, determacy ad falsty membershp fuctos for obectves are W, W 1 f W L U W W f L W U 0 f W U W W 1 exp W W W UW L W W 1 f W L L W W W I W exp W f LW W LW W 0 f W LW W W (6) (63) LW 0 f W W 1 1 U L W W W tah F W W W W f LW W W UW 1 f W UW where 0, U L W W W W (64) ad 1 f L U 1 exp f L U U L 0 f U (65) 143

145 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag 1 f L L I exp f L L 0 f L (66) 0 f L 1 1 U L tah F f L U 1 f U where, for where 0, U L (67) are o-zero parameters prescrbed by the decso maker ad ccordg to eutrosophc optmzato techque cosderg truth, determacy ad falsty membershp fucto for MOSOP (1), crsp o-lear programmg problem ca be formulated as Maxmze Subect to W ; (68) W (69) ; (70) W ; I W (71) ; I (7) W ; F W (73) ; F (74) ; (75) 3; ; ; (76),, 0,1, (77) m max (78) whch s reduced to equvalet o lear programmg problem as Maxmze Such that (79) 144

146 Neutrosophc Operatoal Research Volume II UW LW W UW ; (80) W L W W W ; (81) W W U L ; (8) W W W U L U ; (83) L ; (84) U L W W ; (85) 3; (86) ; ; (87),, 0,1 (88) where l 1 ; (89) 4; (90) 6 W F F UW LW U F l ; 6 L F ; ; (91) (9) (93) 1 tah 1. (94) Solvg the above crsp model (79-94) by a approprate mathematcal programmg algorthm we get optmal soluto ad hece obectve fuctos.e structural weght ad deflecto of the loaded ot wll atta Pareto optmal soluto. 6 Numercal Illustrato well kow three bar plaer truss s cosdered to mmze weght of the structure W 1, ad mmze the deflecto 1, at a loadg pot of a statstcally loaded three bar plaer truss subect to stress costrats o each of the truss members 145

147 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Fg. 1. Desg of the three-bar plaar truss he mult-obectve optmzato problem ca be stated as follows 1, 1 Mmze W L (95) Mmze Subect to, 1 PL E 1 P, ; (96) (97) P, ; 1 1 P, ; C m max (98) (99) 1, (100) P materal desty ; L E Youg s where appled load ; legth ; modulus ; 1 Cross secto of bar-1 ad bar-3; Cross secto of bar-; s deflecto of loaded ot. 1 ad are maxmum allowable tesle C stress for bar 1 ad bar respectvely, 3 s maxmum allowable compressve stress for bar

148 Neutrosophc Operatoal Research Volume II able 1: Iput data for crsp model (79-94) ppled load P KN 0 Volume desty 3 KN / m 100 Legth L 1 m Maxmum allowable tesle stress 1 KN / m 0 Maxmum allowable compressve stress C 3 KN / m 15 Youg s modulus E KN / m 7 10 m ad max of cross secto of bars 10 m m max 1 5 m 0.1 max 5 Soluto: ccordg to step of 4, pay-off matrx s formulated as follows U F W 1,, W Here U , (101) W F L L ; L W I W (10) W 1 1 L , (103) W I U L W (104) W 1 1 such that 0, U F ; 1 1 U , (105) F L L ; L I (106) L , (107) I U L (108) such that 0, Here truth, determacy, ad falsty membershp fucto for obectve fuctos W,,, are defed as follows

149 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag 1 f W 1, W 1, W 1, 1, 1 exp 4 f W 1, W f W 1, (109) 1 f W 1, W 1, I W 1, 1, exp f W 1, W f W 1, (110) 0 f W 1, F W 1, 1, tah 1, , W W f W f W 1, (111) 0, ad f 1, , 1, 1, 1 exp 4 f , f 1, (11) (114) 1 f 1, , I 1, 1, exp f 130 1, f 1, (113) 0 f 1, F 1, 1, tah 1, f , f 1, ,

150 Neutrosophc Operatoal Research Volume II ccordg to eutrosophc optmzato techque the MOSOP (95-100) ca be formulated as Maxmze Such that 1 (115) ; (116) (117) ; ; (118) ; ; 0; ; (119) (10) (11) (1) 0 0; ; (13) (14) 3; (15) ; (16) 1 (17) (18) Now, usg above metoed truth, determacy ad falsty membershp fucto NLP (95-100) ca be solved by NSO techque for dfferet values of, ad 1,. he optmum soluto of MOSOP(95-100) s gve table ()

151 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag able : Comparso of Optmal soluto of MOSOP (95-100) based o dfferet method Methods Neutosophc optmzato (NSO) wth lear membershp fucto , , Neutosophc optmzato (NSO) wth olear membershp fucto , , m 4 10 m 4 W, 1 10 KN, m Here we get same solutos for the dfferet tolerace ad for determacy membershp fucto of obectve fuctos. From the table, t shows that NSO techque gves same Pareto optmal result for lear ad olear membershp fuctos the perspectve of Structural Optmzato. 7 Cocluso he ma obectve of ths work s to llustrate how eutrosophc optmzato techque ca be utlzed to solve a olear structural problem. he cocept of eutrosophc optmzato techque allows oe to defe a degree of truth membershp, whch s ot a complemet of degree of falsty; rather, they are depedet wth degree of determacy. I ths problem, actually, we vestgate the effect of o-lear truth, determacy ad falsty membershp fucto of euotrosophc set perspectve of mult-obectve structural optmzato. Here we have cosdered a o-lear three bar truss desg problem. I ths test problem, we fd out mmum weght of the structure as well as mmum deflecto of loaded ot. he comparsos of results obtaed for the udertake problem clearly show the superorty of eutrosophc optmzato over fuzzy optmzato. he results of ths study may lead to the developmet of effectve eutrosophc techque for solvg other model of olear programmg problem dfferet feld. ckowledgemet he research work of MrdulaSarkar s faced by Rav Gadh Natoal Fellowshp (F1-17.1/ SC-wes-4549/(S-III/Webste)), Govt of Ida. 1, 3 150

152 Neutrosophc Operatoal Research Volume II Refereces [1] taassov, K.. Itutostc fuzzy sets. Fuzzy Sets ad Systems1986:0(1): [] Zadeh, L.. Fuzzy set. Iformato ad Cotrol.1965: 8(3): [3] Rao, S. S. Descrpto ad optmum desg of fuzzy mechacal systems. Joural of Mechasms, rasmssos, ad utomato Desg.1987: 109(1): [4] Bellma, R. E., &Zadeh, L.. Decso-makg a fuzzy evromet. Maagemet scece,1970:17(4): B-141. [5] Shh, C. J., Ch, C. C., & Hsao, J. H. lteratve α-level-cuts methods for optmum structural desg wth fuzzy resources. Computers & structures.003: 81(8): [6] Shh, C. J., & Lee, H. W. Level-cut approaches of frst ad secod kd for uque soluto desg fuzzy egeerg optmzato problems. amkag Joural of Scece ad Egeerg.004:7(3): [7] Smaradache, F. (1998). Neutrosophy, eutrosophc probablty, set ad logc, mer. Res. Press, Rehoboth, US, 105. [8] Wag, G.Y. & Wag, W.Q. Fuzzy optmum desg of structure. Egeerg Optmzato.1985:8: [9] Yeh, Y.C. & Hsu, D.S. Structural optmzato wth fuzzy parameters. Computer ad Structure.1990:37(6): [10] Chagwe, X. Fuzzy optmzato of structures by the two-phase method. Computers & Structures.1989:31(4): [11] Dey, S., & Roy,. K. Fuzzy programmg echque for Solvg Multobectve Structural Problem. Iteratoal Joural of Egeerg ad Maufacturg.014: 4(5): 4. [1] Dey, S., & Roy,. K. Optmzed soluto of two bar truss desg usg tutostc fuzzy optmzato techque. Iteratoal Joural of Iformato Egeerg ad Electroc Busess.014: 6(4): 45. [13] Dey, S., & Roy,. K. Mult-obectve structural desg problem optmzato usg parameterzed t-orm based fuzzy optmzato programmg echque. Joural of Itellget ad Fuzzy Systems.016:30(): [14] Dey, S., & Roy,. Optmum shape desg of structural model wth mprecse coeffcet by parametrc geometrc programmg. Decso Scece Letters.015:4(3): [15] Jaa, B., & Roy,. K. Mult-obectve tutostc fuzzy lear programmg ad ts applcato trasportato model. Notes o Itutostc Fuzzy Sets.007: 13(1): [16] Dey, S., & Roy,. K. Mult-obectve structural optmzato usg fuzzy ad tutostc fuzzy optmzato techque. Iteratoal Joural of Itellget systems ad applcatos.015:7(5):57. [17] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [18] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [19] bdel-basset, Mohamed, Ma Mohamed, ad ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [0] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers:

153 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [1] Mrdula Sarkar, Samr Dey, apa Kumar Roy: russ Desg Optmzato usg Neutrosophc Optmzato echque, Neutrosophc Sets ad Systems, vol. 13, 016, pp do.org/10.581/zeodo [] Ptu Das, apa Kumar Roy: Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem, Neutrosophc Sets ad Systems, vol. 9, 015, pp do.org/10.581/zeodo [3] Moa Gamal Gafar, Ibrahm El-Heawy: Itegrated Framework of Optmzato echque ad Iformato heory Measures for Modelg Neutrosophc Varables, Neutrosophc Sets ad Systems, vol. 15, 017, pp do.org/10.581/zeodo [4] Naga Rau I, Raeswara Reddy P, Dr. Dwakar Reddy V, Dr. Krshaah G: Real Lfe Decso Optmzato Model, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [5] Wezhog Jag, Ju Ye: Optmal Desg of russ Structures Usg a Neutrosophc Number Optmzato Model uder a Idetermate Evromet, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [6] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [7] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [8] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [9] bdel-baset, Mohamed, Ibrahm M. Hezam, ad Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [30] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [31] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016):

154 Neutrosophc Operatoal Research Volume II VIII Mult-Obectve Welded Beam Optmzato usg Neutrosophc Goal Programmg echque Mrdula Sarkar 1 apa Kumar Roy 1Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah , Ida. Emal: mrdula.sarkar86@redffmal.com Ida Isttute of Egeerg Scece ad echology, Shbpur, P.O-Botac Garde, Howrah , Ida. Emal: roy_t_k@yahoo.co. bstract hs paper vestgates mult obectve Neutrosophc Goal Optmzato (NSGO) approach to optmze the cost of weldg ad deflecto at the tp of a welded steel beam, whle the maxmum shear stress the weld group, maxmum bedg stress the beam, ad bucklg load of the beam have bee cosdered as costrats. he problem of desgg a optmal welded beam cossts of dmesog a welded steel beam ad the weldg legth so as to mmze ts cost, subect to the costrats as stated above. he classcal welded bream desg structure s preseted here to demostrate the effcecy of the eutrosophc goal programmg approach. he model s umercally llustrated by geeralzed NSGO techque wth dfferet aggregato method. he result shows that the Neutrosophc Goal Optmzato techque s very effcet fdg the best optmal solutos. Keywords Neutrosophc Set; Sgle Valued Neutrosophc Set; Geeralzed Neutrosophc Goal Programmg; rthmetc ggregato; Geometrc ggregato; Welded Beam Desg Optmzato. 1 Itroducto Weldg, a process of og metallc parts wth the applcato of heat or pressure or the both, wth or wthout added materal, s a ecoomcal ad effcet method for obtag permaet ots the metallc parts. hs welded 153

155 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag ots are geerally used as a substtute for rveted ot or ca be used as a alteratve method for castg or forgg. he weldg processes ca broadly be classfed to followg two groups, the weldg process that uses heat aloe to o two metallc parts ad the weldg process that uses a combato of heat ad pressure for og (Bhadar. V. B). However, above all the desg of welded beam should preferably be ecoomcal ad durable oe. Sce decades, determstc optmzato has bee wdely used practce for optmzg welded coecto desg. hese clude mathematcal optmzato algorthms (Ragsdell & Phllps 1976) such as PPROX (Grffth & Stewart s) successve lear approxmato, DVID (Davdo Fletcher Powell wth a pealty fucto), SIMPLEX (Smplex method wth a pealty fucto), ad RNDOM (Rchardso s radom method) algorthms, G-based methods (Deb 1991, Deb 000, Coello 000b, Coello 008), partcle swarm optmzato (Reddy 007), harmoy search method (Lee & Geem 005), ad Bg-Bag Bg-Cruch (BB- BC) (O. Hasaçeb, 011) algorthm. SOP (O. Hasaçeb, 01), subset smulato (L 010), mproved harmoy search algorthm (Mahadav 007), were other methods used to solve ths problem. Recetly a robust ad relable H statc output feedback (SOF) cotrol for olear systems (Yalg We 016) ad for cotuous-tme olear stochastc systems (Yalg We 016) wth actuator fault a descrptor system framework have bee studed. ll these determstc optmzatos am to search the optmum soluto uder gve costrats wthout cosderato of ucertates. So, whle a determstc optmzato approach s uable to hadle structural performaces such as mprecse stresses ad deflecto etc. due to the presece of ucertates, to get rd of such problem fuzzy (Zadeh, 1965), tutostc fuzzy (taassov,1986), Neutrosophc [7,1-30] play great roles.radtoally structural desg optmzato s a well kow cocept ad may stuatos t s treated as sgle obectve form, where the obectve s kow the weght or cost fucto. he exteso of ths s the optmzato where oe or more costrats are smultaeously satsfed ext to the mmzato of the weght or cost fucto. hs does ot always hold good real world problems where multple ad coflctg obectves frequetly exst. I ths cosequece a methodology kow as mult-obectve optmzato (MOSO) s troducedso to deal wth dfferet mprecseess such as stresses ad deflecto wth multple obectve, we have bee motvated to corporate the cocept of eutrosophc set ths problem, ad have developed mult-obectve eutrosophc optmzato algorthm to optmze the optmum desg.usually Itutostc fuzzy set, whch s the geeralzato of fuzzy sets, cosders both truth membershp ad falsty membershp that ca hadle complete formato excludg the determate ad cosstet formato whle eutrosophc set ca quatfy determacy explctly by defg truth, determacy ad falsty membershp fucto 154

156 Neutrosophc Operatoal Research Volume II depedetly. herefore, Wag et.al (010) preseted such set as sgle valued eutrosophc set (SVNS) as t comprsed of geeralzed classc set, fuzzy set, terval valued fuzzy set, tutostc fuzzy set ad Para-cosstet set. s applcato of SVNS optmzato method s rare welded beam desg, hece t s used to mmze the cost of weldg by cosderg shear stress, bedg stress the beam, the bucklg load o the bar, the deflecto of the beam as costrats. herefore the result has bee compared amog three cted methods each of whch mprecseess has bee cosdered completely dfferet way.moreover usg above cted cocept, a mult-obectve eutrosophc optmzato algorthm has bee developed to optmze three bar truss desg (Sarkar 016), ad to optmze rser desg problem (Das 015). I early 1961 Chares ad Cooper frst troduced Goal programmg problem for a lear model. Usually coflctg goal are preseted a mult-obectve goal programmg problem. Dey et al.(015) used tutostc goal programmg o olear structural model.however, the factors goverg of former costrats are heght ad legth of the welded beam, forces o the beam, momet of load about the cetre of gravty of the weld group, polar momet of erta of the weld group respectvely. Whle, the secod costrat cosders forces o the beam, legth ad sze of the weld, depth ad wdth of the welded beam respectvely. hrd costrat cludes heght ad wdth of the welded beam. Fourth costrats cossts of heght, legth, depth ad wdth of the welded beam. Lastly ffth costrat cludes heght of the welded beam. Besdes, flexblty has bee gve shear stress, bedg stress ad deflecto oly, hece all these parameters become mprecse ature so that t ca be cosdered as eutrosophc set to from truth, determacy ad falsty membershp fuctos Ultmately, eutrosophc optmzato techque has bee appled o the bass of the cted membershp fuctos ad outcome of such process provdes the mmum cost of weldg,mmum deflecto for olear welded beam desg. he comparso of results shows dfferece betwee the optmum value whe partally ukow formato s fully cosdered or ot. hs s the frst tme NSGO techque s applcato to mult-obectve welded beam desg. he preset study vestgates computatoal algorthm for solvg multobectve welded beam problem by sgle valued geeralzed NSGO techque. he results are compared umercally for dfferet aggregato method of NSGO techque. From our umercal result, t has bee see that the best result obtaed for geometrc aggregato method for NSGO techque the perspectve of structural optmzato techque. 155

157 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Mult-obectve Structural Model I szg optmzato problems, the am s to mmze mult obectve fucto, usually the cost of the structure, deflecto uder certa behavoural costrats whch are dsplacemet or stresses. he desg varables are most frequetly chose to be dmesos of the heght, legth, depth ad wdth of the structures. Due to fabrcatos lmtatos the desg varables are ot cotuous but dscrete for beloggess of cross-sectos to a certa set. dscrete structural optmzato problem ca be formulated the followg form (P1) Mmze C( X ) (1) Mmze X (), 1,,..., subect to X X m (3) X R d, 1,,..., (4) where ( ), X ad X as represet cost fucto, deflecto ad the C X behavoural costrats respectvely whereas X deotes the maxmum allowable value, m ad are the umber of costrats ad desg varables respectvely. gve set of dscrete value s expressed by obectve fuctos are take as m t ad X C X c x t t1 1 ad costrat are chose to be stress of structures as follows c t wth allowable tolerace 0 for 1,,..., m where s the cost coeffcet of t th sde ad s the respectvely, m s the umber of structural elemet, ad allowable stress respectvely 3 Mathematcal Prelmares x d R th 0 ad ths paper desg varable are the th stress, I the followg, we brefly descrbe some basc cocepts ad basc operatoal laws related to eutrosophc set 3.1 Fuzzy Set (FS) Let X be a fxed set. fuzzy set of X s a obect havg the form 156

158 Neutrosophc Operatoal Research Volume II x, x x X where the fucto : 0,1 x X (5) to the set. 3. Itutostc Fuzzy Set(IFS) X stads for the truth membershp of the elemet Let a set X be fxed. tutostc fuzzy set or IFS X s a obect of the form X, x, F x x X (6) where : X 0,1 ad : 0,1 F X defe the truth membershp ad falsty membershp respectvely, for every elemet of 0 x F x Sgle-Valued Neutrosophc Set (SVNS) x X such that Let a set X be the uverse of dscourse. sgle valued eutrosophc set over X s a obect havg the form,,, (7) x x I x F x x X where I X ad : 0,1 ad falsty membershp fuctos respectvely so as to x I x F x : X 0,1, : 0,1 for all x X. 3.4 Uo of Neutrosophc Sets (NSs) he uo of two sgle valued eutrosophc sets valued eutrosophc set U deoted by F X are truth, determacy 0 3 ad B s a sgle U B x, x, I x, F x x X (8) U U U ad s defed by the followg codtos () x x x max,, U B () I x I x I x max,, U B () F x m F x, F x for all x X U B for ype-i Or aother way by defg followg codtos 157

159 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag () x x x max,, U B () I x m I x, I x U B () F x m F x, F x for all x X U B for ype-ii where U, x I U x, F U x represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be calculated for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.6, 0.4, 0. / x 0.5, 0., 0.3 / x 0.7, 0., 0. / x (9) ad for ype -II as 1 3 B 0.6, 0.1, 0. / x 0.5, 0., 0.3 / x 0.7, 0.1, 0. / x (10) 3.5 Itersecto of Neutrosophc Sets 1 3 he tersecto of two sgle valued eutrosophc sets sgle valued eutrosophc set E s deoted by ad B s a,,, (11) E B x x I x F x x X E E E ad s defed by the followg codtos () x x x m,, E B () I x I x I x m,, E B () F x max F x, F x for all x X E B for ype-i Or aother way by defg followg codtos () x x x m,, E B () I x max I x, I x E B 158

160 Neutrosophc Operatoal Research Volume II () F x max F x, F x for all x X E B for ype-ii where E x, I E x, F E x represet truth membershp, determacymembershp ad falsty-membershp fuctos of uo of eutrosophc sets Example: Let 0.3, 0.4, 0.5 / x1 0.5, 0., 0.3 / x 0.7, 0., 0. / x3 ad B 0.6, 0.1, 0. / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x be two eutrosophc sets. he the uo of set ca be measured for ype -I as 1 3 ad B s a sgle valued eutrosophc B 0.3, 0.1, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0.1, 0.5 / x (1) ad for ype -II as 1 3 B 0.3, 0.4, 0.5 / x 0.3, 0., 0.6 / x 0.4, 0., 0.5 / x (13) Mathematcal alyss 4.1 Neutrosophc Goal Programmg Goal programmg ca be wrtte as (P) to acheve: Fd,,..., x x1 x x (14) z t 1,,..., k (15) Subect to x X where t are scalars ad represet the target achevemet levels of the obectve fuctos that the decso maker wshes to atta provded, X s feasble set of costrats. he olear goal programmg problem ca be wrtte as F,,..., x x1 x x So as to (16) tolerace Mmze z wth target value d reecto tolerace c t,acceptace tolerace a,determacy 159

161 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag x X (17), 1,,..., m (18) g x b x 0, 1,,..., falsty-membershp fuctos wth truth-membershp, determacy-membershp ad 1 f z t t a z z f t z t a 0 f z t a I 1 a 1 z 0 f z t z t f t z t a d t a z f t d z t a a d 0 f z t a 0 f z t z t F z f t z t c 1 f z t c 1 c (19) (0) (1) o maxmze the degree of acceptace ad determacy of olear goal programmg (NGP) obectves ad costrats also to mmze degree of reecto of of NGP obectves ad costrats, (P3) Maxmze z, 1,,..., k () z Maxmze I z, 1,,..., k (3) z Mmze F z, 1,,..., k (4) Subect to z 0 z I z F z 3, 1,,..., k (5) z z z z 0, I z 0, F z I 1,,..., k (6) z z z 160

162 Neutrosophc Operatoal Research Volume II where, 1,,..., z I z I k (7) z z, 1,,..., z F z k (8) z z g x b x 0, 1,,..., z z, I, 1,,..., m (9) z z ad F z z (30) are truth membershp fucto determacy membershp fucto,falsty membershp fucto of eutrosophc decso set respectvely. Now the eutrosophc goal programmg (NGP) model (P3) ca be represeted by crsp programmg model usg truth membershp, determacy membershp,ad falsty membershp fuctos as (P4) Maxmze, Maxmze, Mmze (31) z, 1,,..., z k (3) z, 1,,..., I z k (33) z, 1,,..., F z k (34) z t, 1,,..., k (35) 0 3; (36), 0, 1; (37), 1,,..., (38) g x b m x 0, 1,,..., (39) 4. Geeralzed Neutrosophc Goal Programmg he geeralzed eutrosophc goal programmg ca be formulated as (P5) Maxmze z, 1,,..., k (40) z Maxmze I z, 1,,..., k (41) z 161

163 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Mmze F z, 1,,..., k (4) Subect to z z I z F z w w w, 1,,..., k (43) z z z z 0, I z 0, F z I 1,,..., k (44) z z z, 1,,..., z I z I k (45) z z, 1,,..., z F z k (46) z z 0 w w w 3 (47) w, w, w 0,1 (48) g x b, 1,,..., m (49) x 0, 1,,..., (50) Equvaletly (P6) Maxmze, Maxmze, Mmze (51) z, 1,,..., z k (5) z, 1,,..., I z k (53) z, 1,,..., F z k (54) z t, 1,,..., k (55) 0 w w w ; (56) 1 3 0, w1, 0, w, 0, w3 ; (57) 0,1, 0,1, 0,1 ; (58) w w w w w w 3; (59) 1 3, 1,,..., (60) g x b m x 0, 1,,..., (61) 16

164 Neutrosophc Operatoal Research Volume II Equvaletly (P7) Maxmze, Maxmze, Mmze (6) z t a 1, 1,,..., k w1 (63) d z t, 1,,..., k w (64) z t a a d, 1,,..., k (65) w 3 c z t, 1,,..., k w (66) z t, 1,,..., k (67) 0 w w w ; (68) 1 3 0, w1, 0, w, 0, w3 ; (69) 0,1, 0,1, 0,1 ; (70) w w w w w w 3; (71) 1 3 Wth the help of geeralzed truth, determacy, falsty membershp fucto the geeralzed eutrosophc goal programmg based o arthmetc aggregato operator ca be formulated as (P8) Mmze (7) Subect bect to the same costrats as (P7) Wth the help of geeralzed truth, determacy, falsty membershp fucto the geeralzed eutrosophc goal programmg based o geometrc aggregato operator ca be formulated as (P9) Mmze (73) Subected to same costrats as (P7) 163

165 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Now ths o-lear programmg problem (P7 or P8 or P9) ca be easly solved by a approprate mathematcal programmg to gve soluto of multobectve o-lear programmg problem (P1) by geeralzed eutrosophc goal optmzato approach. 5 Soluto of Mult-obectve Welded Beam Optmzato Problem (MOWBP) by Geeralzed Neutrosophc Goal Optmzato echque as he mult-obectve eutrosophc fuzzy structural model ca be expressed (P10) tolerace Mmze C X wth target value d C ad reecto tolerace c C C 0,truth tolerace a C,determacy (74) tolerace Mmze d 0 X wth target value ad reecto tolerace c 0 0,truth tolerace a 0,determacy (75) subect to x x x m max X (76) (77) X x1 x x where,,..., (78) are the desg varables, s the group umber of desg varables for the welded beam desg. o solve ths problem we frst calculate truth, determacy ad falsty membershp fucto of obectve as follows w1 f C X C0 w C 1 0 ac C X C C X w1 f C0 C X C0 a a C 0 f C X C0 ac C (79) 164

166 Neutrosophc Operatoal Research Volume II I C X w C X 0 f C X C0 C X C 0 w f C0 C X C0 a C d C C0 ac C X w f C0 dc C X C0 a ac d C 0 f C X C0 ac C (80) where d C w1 w w a c 1 C C (81) 0 f C X C0 w C X C 3 0 FC X C X w3 f C0 C X C0 c c C w3 f C X C0 cc C (8) d w1 f X 0 0 a w X 1 0 X X w1 f 0 X 0 a a 0 0 f X 0 a 0 I X X w 0 f X 0 X 0 w f 0 X 0 a d 0 a W X w f 0 d X 0 a a d 0 f X 0 a 0 (83) (84) d w1 w w a c 1 0 f X 0 w X 3 0 F X X w3 f 0 X 0 c c w3 f X 0 c (85) 165

167 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag ccordg to geeralzed eutrosophc goal optmzato techque usg truth, determacy ad falsty membershp fucto, MOSOP (P1) ca be formulated as Model -I Maxmze, Maxmze, Mmze (86) C X C0 a C 1, w1 (87) C0 C X d C, w (88) C X C a a d, (89) 0 C0 C X C C C w c C, w (90) 3 C, (91) 0 C X a 1, w1 X 0 (9) d X 0, (93) w, (94) X a a d 0 w c X 0, (95) w 3 X, (96) 0 0 w w w ; (97) 1 3 0, w1, 0, w, 0, w3 ; (98) 0,1, 0,1, 0,1 ; (99) w w w w w w 3; (100) 1 3, 1,,..., (101) X m x 0, 1,,..., (10) 166

168 Neutrosophc Operatoal Research Volume II Wth the help of geeralzed truth, determacy, falsty membershp fucto the geeralzed eutrosophc goal programmg based o arthmetc aggregato operator ca be formulated as: Model -II Mmze (103) Subected to same costrat as Model I Wth the help of geeralzed truth, determacy, falsty membershp fucto the geeralzed eutrosophc goal programmg based o geometrc aggregato operator ca be formulated as Model -III Mmze (104) Subected to same costrat as Model I Now these o-lear programmg Model-I,II,III ca be easly solved through a approprate mathematcal programmg to gve soluto of multobectve o-lear programmg problem (8) by geeralzed eutrosophc goal optmzato approach. 6 Numercal Illustrato Weldg, a process of og metallc parts wth the applcato of heat or pressure or the both, wth or wthout added materal, s a ecoomcal ad effcet method for obtag permaet ots the metallc parts. hs welded ots are geerally used as a substtute for rveted ot or ca be used as a alteratve method for castg or forgg. he weldg processes ca broadly be classfed to followg two groups, the weldg process that uses heat aloe to o two metallc parts ad the weldg process that uses a combato of heat ad pressure for og (Bhadar. V. B). However, above all the desg of welded beam should preferably be ecoomcal ad durable oe. 6.1 WBD Formulato he optmum welded beam desg(fg. 1) ca be formulated cosderg some desg crtera such as cost of weldg.e cost fucto, shear stress, bedg stress ad deflecto, derved as follows: 167

169 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Cost Fucto Formulato he performace dex approprate to ths desg s the cost of weld assembly. he maor cost compoets of such a assembly are () set up labour cost, () weldg labour cost, () materal cost,.e where, C X C0 C1 C (105) C X materal cost. Now: cost fucto; C0 set up cost; C1 weldg labour cost; C Set up cost C 0 : he compay has chose to make ths compoet a weldmet, because of the exstece of a weldg assembly le. Furthermore, assume that fxtures for set up ad holdg of the bar durg weldg are readly avalable. he cost ca therefore be gored ths partcular total cost model. C 0 Weldg labour cost C 1 : ssume that the weldg wll be doe by mache at a total cost of $10/hr (cludg operatg ad mateace expese). Furthermore suppose that the mache ca lay dow a cubc ch of weld 6 m. he labour cost s the $ 1 $ m $ C V 1 3 w V 3 hr 60 m w (106) Where Vw weld volume, 3 Materal cost C : C C3V w C4VB (107) Where C3 cost per volume per weld materal,$/ 3 (0.37)(0.83) ; cost per volume of bar stock,$/ 3 (0.37)(0.83) ; B VB volume of bar, 3. C4 From geometry V h l ; volume of the weld materal, 3 ; V x x ad w ; volume of bar, 3 V x x L x V tb L l bar herefore cost fucto become C X h l C h l C tb L l x x x x x (108) weld 1 168

170 Neutrosophc Operatoal Research Volume II Costrats Dervato from Egeerg Relatoshp Fg. 1 Shear stresses the weld group. Maxmum shear stress weld group o complete the model t s ecessary to defe mportat stress states Drect or prmary shear stress.e Load P P P 1 (109) hroat area hl x x Sce the shear stress produced due to turg momet M P. e at ay secto, s proportoal to ts radal dstace from cetre of gravty of the ot G, therefore stress due to M s proportoal to ad s a drecto at rght agles to. I other words costat (110) R r R 1 R herefore x x x l h t R (111) Where, s the shear stress at the maxmum dstace R ad s the shear stress at ay dstace r. Cosder a small secto of the weld havg area d a dstace r from G. herefore shear force o ths small secto turg momet of the shear force about cetre of gravty s dm d r d r R at d ad (11) herefore total turg momet over the whole weld area 169

171 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag where M d r J. R (113) R J polar momet of erta of the weld group about cetre of gravty. herefore, shear stress due to the turg momet.e. secodary shear stress, MR J (114) I order to fd the resultat stress, the prmary ad secodary shear stresses are combed vectorally. herefore the maxmum resultat shear stress that wll be produced at the weld group, cos, (115) 1 1 where, agle betwee 1 ad. s cos l x ; R R (116) x 1 1. (117) R Now the polar momet of erta of the throat area gravty s obtaed by parallel axs theorem, about the cetre of x x l l x 1 3 J I xx x x x x1x (118) Where, throat area x1x, l Legth of the weld, x Perpedcular dstace betwee two parallel axes t h x x 1 3 (119) Maxmum bedg stress beam Now Maxmum bedg momet where PL; PL, maxmum bedg stress, Z 3 I bt Z secto modulus ; I momet of erta ; y dstace of y 1 t bt extreme fbre from cetre of gravty of cross secto ; herefore Z

172 Neutrosophc Operatoal Research Volume II 6PL 6PL Z bt x x So bar bedg stress x. (10) 4 3 Maxmum deflecto beam Maxmum deflecto at catlever tp PL PL 4PL 4PL x 3 3 3EI bt Ebt Ex4x (11) 3 3E 1 Bucklg load of beam Bucklg load ca be approxmated by P C l x EIC a El 1 l C (1) 6 tb E 36 L t E EGx3x 4 /36 x3 E 1 L 4 G 1 L L 4 G (13) where, I momet of erta 3 bt ; torsoal rgdty 1 1 t ; l L; a. 3 3 C GJ tb G 6. Crsp Formulato of WBD I desg formulato a welded beam (Fg. ) has to be desged at mmum cost whose costrats are shear stress weld,bedg stress the beam,bucklg load o the bar P,ad deflecto of the beam.he x1 h x l desg varables are where h s the weld sze, s the legth of the weld x 3 t x4 b, t s the depth of the welded beam, b s the wdth of the welded beam. l 171

173 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Fg. Desg of the welded beam he sgle-obectve crsp welded beam optmzato problem ca be formulated as follows: R P C (P11) (14) Mmze C X x x x x x such that (15) g x x 1 max 0 (16) g x x max 0 (17) g x x x g x x x x x 14 x 5 0 (18) (19) g x x 5 1 (130) g x x 6 max 0 (131) g x P P x 7 C x, x, x, x 0,1 (13) x R where x x x1 x3 x 1 1 ; 1 4 ; x1 x x x1 x3 J ; 1 EGx x x E L G / L 4 P MR ; x ; M P L xx J ; 1 6PL xx x ; x PL ; Ex x 4 3 as derved as Eq.(109), Eq.(114), Eq.(113), Eq.(111), Eq.(118), Eq.(10), Eq.(11),, Eq.(13), respectvely. ga P Force o beam ; L Beam legth beyod weld; x1 Heght of the welded 17

174 Neutrosophc Operatoal Research Volume II beam; x Legth of the welded beam; x3 Depth of the welded beam; x4 Wdth of the welded beam; x Desg shear stress; x Desg ormal stress for beam materal; M Momet of about the cetre of gravty of the weld, Shearg modulus of J Polar momet of erta of weld group; Beam Materal; E Youg modulus; max Desg ormal stress for the beam materal; Prmary stress o weld throat, P G Desg Stress of the weld; max Maxmum deflecto; Secodary torsoal stress o weld. max 1 able 1: Iput data for crsp model (P11) ppled load P lb 6000 Beam legth beyod weld L 14 Youg Modulus E ps Value of G ps Maxmum allowable shear stress max ps wth fuzzy rego 50 Maxmum allowable ormal stress ps max wth fuzzy rego 50 Maxmum allowable deflecto max 0.5 wth fuzzy rego 0.05 hs mult obectve structural model ca be expressed as eutrosophc fuzzy model as wth target value Mmze C X x x x x x tolerace (133) 5 Mmze w1,determacy tolerace 0.w 0.14w 3 4PL Ex x ad reecto tolerace,truth x ; wth target value 0.0,truth tolerace w1,determacy tolerace 4.34w 4.16w Subect to 1 ad reecto tolerace 0.4 (134) (135) g x x 1 max 0; 7 173

175 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag (136) g x x max 0; (137) g x x x ; g x x x x x 14 x 5 0; (138) ; (139) g x x 5 1 (140) g x x 6 max 0; (141) g7 x P P x 0; 1 4 C 0.1 x, x.0 (14) 0.1 x, x.0 (143) 3 x 1 1 ; R where x 1 P xx 1 ; MR J x M P L ; ; R P x x1 x3 C x 4 ; x1 x x x1 x3 J ; 1 L 4G EGx3x 4 / 36 x3 E 1 ; L 6PL xx x ; x PL ; Ex x 4 3 ccordg to geeralzed eutrosophc goal optmzato techque usg truth, determacy ad falsty membershp fucto, MOWBP (P11) ca be formulated as Model -I Maxmze, Maxmze, Mmze (144) x1 x x x3x , w1 w x x x x x 3.39, w w w w x1 x x x3x , w 0.w1 0.14w (145) (146) (147) x 1 x x x3x , w (148) (149) x x x x x 3.39,

176 Neutrosophc Operatoal Research Volume II 3 4PL Ex x PL Ex x , w1 w w 4.3w 4.1w 1 0.0, 1 (150) (151) 3 4PL Ex x 4 3 w w 4.3w 4.1w , 1 (15) R 3 4PL 0.4 Ex x , (153) w 3 3 4PL 0.0, Ex x (154) 0 w w w ; (155) 1 3 0, w1, 0, w, 0, w3 ; (156) 0,1, 0,1, 0,1 ; (157) w w w w w w 3; (158) 1 3 (159) g x x 1 max 0; (160) g x x max 0; (161) g x x x ; g x x x x x 14 x 5 0; (16) ; (163) g x x 5 1 (164) g x x 6 max 0; (165) g7 x P P x 0; 1 4 C 0.1 x, x.0 (166) 0.1 x, x.0 (167) 3 x R where x x x1 x3 1 1 ; 1 4 ; x1 x x x1 x3 J ; 1 P MR ; x ; M P L xx J ; 1 6PL xx x ; x PL ; Ex x

177 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag P C x L 4G EGx3x 4 / 36 x3 E 1 ; L Wth the help of geeralzed truth, determacy, falsty membershp fucto the geeralzed eutrosophc goal programmg problem (P11)based o arthmetc aggregato operator ca be formulated as Model -II Mmze (168) subected to same costrats as Model-I Wth the help of geeralzed truth, determacy, falsty membershp fucto the geeralzed eutrosophc goal programmg problem (10) based o geometrc aggregato operator ca be formulated as Model -III Mmze (169) subected to same costrats as Model-I Now these o-lear programmg problem Model-I,II,III ca be easly solved by a approprate mathematcal programmg to gve soluto of multobectve o-lear programmg problem (P11) by geeralzed eutrosophc goal optmzato approach ad the results are show the table 1 s gve table.ga value of membershp fucto GNGP techque for MOWBP (P11) based o dfferet ggregato s gve able 3. able : Comparso of GNGP soluto of MOWBP (9) based o dfferet ggregato Methods x1 x x3 x4 C X X Geeralzed Fuzzy Goal programmg(gfgp) w Geeralzed Itutostc Fuzzy Goal programmg(gifgp) w w

178 Neutrosophc Operatoal Research Volume II Geeralzed Neutrosophc Goal programmg (GNGP) w1 0.4, w 0.3, w3 0.7 Geeralzed Itutostc Fuzzy optmzato (GIFGP) based o rthmetc ggregato w 0.15, w Geeralzed Neutosophc optmzato (GNGP) based o rthmetc ggregato w 0.4, w 0.3, w Geeralzed Itutostc Fuzzy optmzato (GIFGP) based o Geometrc ggregato w 0.15, w Geeralzed Neutosophc optmzato (GNGP) based o Geometrc ggregato w 0.4, w 0.3, w Here we almost same solutos for the dfferet value of w1, w, w 3 dfferet aggregato method for obectve fuctos. From able. t s clear that the cost of weldg ad deflecto are almost same fuzzy ad tutostc fuzzy as well as eutrosophc optmzato techque. Moreover t has bee see that desred value obtaed dfferet aggregato method have ot affected by varato of methods perspectve of welded beam desg optmzato. 7 Cocluso he research study vestgates that eutrosophc goal programmg ca be utlzed to optmze a olear welded beam desg problem. he results obtaed for dfferet aggregato method of the udertake problem show that the best result s acheved usg geometrc aggregato method. he cocept of eutrosophc optmzato techque allows oe to defe a degree of truth 177

179 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag membershp, whch s ot a complemet of degree of falsty; rather, they are depedet wth degree of determacy. s we have cosdered a o-lear welded beam desg problem ad fd out mmum cost of weldg of the structure as well as mmum deflecto, the results of ths study may lead to the developmet of effectve eutrosophc techque for solvg other model of olear programmg problem dfferet feld. ckowledgemet he research work of Mrdula Sarkar s faced by Rav Gadh Natoal Fellowshp (F1-17.1/ SC-wes-4549/(S-III/Webste)), Govt of Ida. Refereces [1] Zadeh, L.. Fuzzy set. Iformato ad Cotrol, 1965:8(3): [] Coello, C..C. 000b. Use of a self-adaptve pealty approach for egeerg optmzato problems. Comput. Id., 41: DOI: /S (99) [3] Reddy, M. J.; Kumar, D. N. effcet mult-obectve optmzato algorthm based o swarm tellgece for egeerg desg. Egeerg Optmzato. 007:39(1) : [4] Carlos. Coello Coello, Solvg Egeerg Optmzato Problems wth the Smple Costraed Partcle Swarm Optmzer, Iformatca.008:3: [5] Lee, K.S., Geem, Z.W. ew meta-heurstc algorthm for cotuous egeerg optmzato: harmoy search theory ad practce Comput. Methods ppl. Mech. Egrg..005:194: [6] S. Kazemzadeh zada, O. Hasaçeba ad O. K. Erol Evaluatg effcecy of bg-bag bg-cruch algorthm bechmark egeerg optmzato problems,it. J. Optm. Cvl Eg., 011:3: [7] Hasaçeb, O. ad zad, S.K. effcet metaheurstc algorthm for egeerg optmzato: SOP t.. optm. cvl eg., 01:(4): [8] Mahdav, M., Fesaghary, M., Damagr, E. mproved harmoy search algorthm for solvg optmzato problems.ppled Mathematcs ad Computato.007:188: [9] taassov, K.. Itutostc fuzzy sets. Fuzzy Sets ad Systems.1986:0(1): [10] Smaradache, F. Neutrosophy, eutrosophc probablty, set ad logc, mer. Res. Press, Rehoboth, US, [11] Shuag L,G. ad u,s.k.solvg costraed optmzato problems va Subset Smulato.010 4th Iteratoal Workshop o Relable Egeerg Computg (REC 010),do: / [1] Yalg We, Jab Qu, Hamd Reza Karm, Relable Output Feedback Cotrol of Dscrete-me Fuzzy ffe Systems wth ctuator Faults. Do: /CSI ,016,1-1. [13] Yalg We, Jab Qu, Hak-Keug Lam, ad Lgag Wu, pproaches to - S Fuzzy-ffe-Model-Based Relable Output Feedback Cotrol for Nolear Itˆo Stochastc Systems.016, DOI /FUZZ ,pp

180 Neutrosophc Operatoal Research Volume II [14] Sarkar, M., Dey, Samr, Roy,.K. Mult-Obectve Neutrosophc Optmzato echque ad ts pplcato to Structural Desg, Iteratoal Joural of Computer pplcatos.016:148(1) :( ). [15] Das, P., Roy,.K. Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem. Neutrosophc Sets ad Systems.015:9: [16] K. Deb, Optmal desg of a welded beam va geetc algorthms, I Joural.1991:9 (11): [17] Deb, K., Pratap,. ad Motra, S., Mechacal compoet desg for multple obectves usg eltst o-domated sortg G. I Proceedgs of the Parallel Problem Solvg from Nature VI Coferece, Pars. 000: 16 0 : [18] K.M. Ragsdell, D.. Phllps, Optmal desg of a class of welded structures usg geometrc programmg, SME Joural of Egeerg for Idustres.1976:98 (3): , Seres B. [19] Chars,., & Cooper, R. Maagemet Models ad Idustral pplcato of Lear Programmg [0] Dey, S., & Roy,. K. Itutostc Fuzzy Goal Programmg echque for Solvg No-Lear Mult-obectve Structural Problem. Joural of Fuzzy Set Valued alyss:015:(3): [1] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [3] bdel-basset, Mohamed, Ma Mohamed, ad ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [4] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [5] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [6] Mrdula Sarkar, Samr Dey, apa Kumar Roy: russ Desg Optmzato usg Neutrosophc Optmzato echque, Neutrosophc Sets ad Systems, vol. 13, 016, pp do.org/10.581/zeodo [7] Ptu Das, apa Kumar Roy: Mult-obectve o-lear programmg problem based o Neutrosophc Optmzato echque ad ts applcato Rser Desg Problem, Neutrosophc Sets ad Systems, vol. 9, 015, pp do.org/10.581/zeodo [8] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [9] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [30] bdel-baset, Mohamed, Ibrahm M. Hezam, ad Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [31] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [3] Moa Gamal Gafar, Ibrahm El-Heawy: Itegrated Framework of Optmzato echque ad Iformato heory Measures for Modelg Neutrosophc Varables, Neutrosophc Sets ad Systems, vol. 15, 017, pp do.org/10.581/zeodo

181 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [33] Naga Rau I, Raeswara Reddy P, Dr. Dwakar Reddy V, Dr. Krshaah G: Real Lfe Decso Optmzato Model, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [34] Wezhog Jag, Ju Ye: Optmal Desg of russ Structures Usg a Neutrosophc Number Optmzato Model uder a Idetermate Evromet, Neutrosophc Sets ad Systems, vol. 14, 016, pp do.org/10.581/zeodo [35] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016): [36] Rttk Roy, Ptu Das: Neutrosophc Goal Programmg appled to Bak - hree Ivestmet Problem, Neutrosophc Sets ad Systems, vol. 1, 016, pp do.org/10.581/zeodo

182 Neutrosophc Operatoal Research Volume II IX Neutrosophc Modules Necat Olgu 1 Mkal Bal 1 Gazatep Uversty, Faculty of Scece ad rt, Departmet of Mathematcs, Gazatep, 7310, urkey. Emal: olgu@gatep.edu.tr Gazatep Uversty, Faculty of Scece ad rt, Departmet of Mathematcs, Gazatep, 7310, urkey. Emal: mkalbal46@hotmal.com bstract hs paper s devoted to the study of eutrosophc modules ad eutrosophc submodules. Neutrosophc logc s a exteso of the fuzzy logc whch determacy s cluded. Neutrosophc Sets are a sgfcat tool of descrbg the completeess, determacy, ad cosstecy of the decso-makg formato. Modules are oe of fudemetal ad rch algebrac structure wth respect to some bary operato the study of algebra. I ths paper, for the frst tme, we study to some basc defto of eutrosophc R-modules ad eutrosophc submodules algebra are geeralzed. Some propertes of eutrosophc R-modules ad eutrosophc submodules are preseted. I ths study, we utlzed classcal modules ad eutrosophc rgs. Cosequetly, we troduced eutrosophc R- modules whch s completely dfferet from the classcal module the structural propertes. I addto, eutrosophc quotet modules ad eutrosophc R-module homomorphsm are explaed ad some deftos, theorems are gve. Fally, some useful examples are gve to verfy the valdty of the proposed deftos ad results. Keywords Neutrosophc group; Neutrosophc rg; Neutrosophc R-module; Weak eutrosophc R-module; Strog eutrosophc R-module; Neutrosophc R-module homomorphsm. 1 Itroducto Neutrosophy s a ew brach of phlosophy, whch studes the ature, org ad scope of eutraltes as well as ther teracto wth deatoal spectra. 181

183 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Neutrosophy s the base of eutrosophc logc whch s a exteso fuzzy logc whch determacy s cluded. Floret Smaradache [6] troduced the Noto of eutrosophy as a ew brach of phlosophy fter, he troduced the cocept of eutrosophc logc ad eutrosophc set where each proposto eutrosophc logc s approxmoted to have the percatage of truth a subset, the percetage of determacy a subset I ad the percatage of falsty a subset F so that ths eutrosophc logc s called a exteso of fuzzy logc especally to tutostc fuzzy logc. I fact eutrosophc set s the geeralzato of classcal set, eutrosophc group ad eutrosophc rg the gearalzato of classcal group ad rg etc. Same way eutrosophc R-module s the geeralzato of classcal R-module. By utlzg the dea of eutrosophc theory Vasatha Kaadasamy ad Floret Samaradache studed eutrosophc algebrac structures by sertg a determate elemet I the algebrc structure ad the combe I wth each elemet of the structure wth respect to correspodg bary operato. hey call t eutrosophc elemet ad the geerated algebrac structure s the termed as eutrosophc algebrac structure. hey further study several eutrosophc algebrac structure such as eutrosophc felds, eutrosophc groups, eutrosophc rgs, eutrosophc semgroups, eutrosophc vector spaces, eutrosophc N-groups, eutrosophc N-semgroups, eutrosophc N loops, eutrosophc bloops, eutrosophc N-loops etc. Groups are so much mportat algebrac structures as they effectve almost all algebrac structures theory. Groups are thought as old algebra due to ts rch structure tha ay other oto. Same way eutrosophc groups are much mportat eutrosophc otos formato. Because they are basc structure almost all eutrosophc otos. Neutrosophc logc has wde applcatos scece, egeerg, poltcs, ecoomcs, etc. herefore, eutrosophc structures are very mportat ad wdely area of study. If reader wated to see detals of eutrosophy ad eutrosophc algebrac structures, the reader should see [3-1]. Prelmares Defto.1: [9] Let (G,*) be ay group ad <G I>={a+bI : a,bg}. N(G)={<G I>,*} s called a eutrosophc group geerated by G ad I uder the bary operato *. I s called the eutrosophc elemet wıth the property I =I. For a teger, +1, ad I are eutrosophc elemets ad 0.I=0.I -1, the verse of I s ot defed ad hece does ot exst. Example 1: (N(Z),+) s a eutrosophc group of tegers ad (N(Q),+) s a eutrosophc group of ratoal umbers. 18

184 Neutrosophc Operatoal Research Volume II Example : Let Z 5 ={0,1,,3,4,5} be a group uder addto modulo 5. N(G)={ <Z 5 I>, + modulo 5} s a eutrosophc group whch s a group. For N(G) ={a+bi a,b Z 5 } s a group uder + modulo 5. hus, ths Neutrosophc group s also a group. heorem.: [1] Let (G,*) be ay group, N(G)={<G I>,*} be the eutrosophc group. (1) N(G) geeral s ot a group. () N(G) always cota a group. Proof: (1) Suppose that N(G) s geeral a group. Let XN(G) be arbtrary. If x s a eutrosophc elemet the X -1 N(G) ad cosequetly N(G) s ot a group, a cotradcto. () Sce a group G ad a determate I gearate N(G), t follows that G N(G) ad N(G) always cota a group. eorem.3. [9] N(G) be ay eutrosophc group. (N(G),*) s commutatve eutrosophc group f a,bn(g), ab=ba. Defto.4. [9] Let N(G) = G I be a eutrosophc group geerated by G ad I. proper subset P(G) s sad to be a eutrosophc subgroup f P(G) s a eutrosophc group.e. P(G) must cota a (sub) group. eorem.5. [1] Let N(M) ad N(P) be ay two eutrosophc subgroups of a commtatve eutrosophc group N(G). (1) N(M) N(P) s a eutrosophc subgroup of N(G). () N(M).N(P) s a eutrosophc subgroup of N(G). (3) N(M) N(P) s a eutrosophc subgroup of N(G) f ad oly f N(M) N(P) or N(P) N(M). Defto.6: [9] eutrosophc group N(G) whch has o otrval eutrosophc ormal subgroups s called a smple eutrosophc group. Defto.7: [] Let (R,+,*) be ay rg. he set <R I>={a+bI a,br} s eutrosophc rg geerated by R ad I uder the operato of R. Example 3: Let Z be the rg of teger <Z I>={a+bI a,bz}. <Z I> s a rg called the eutrosophc rg of teger. lso Z <Z I>. 183

185 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Example 4: Let Q be the rg of ratoal umbers <Q I>={a+bI a,bq}. <Q I> s a rg called the eutrosophc rg of ratoal umbers. lso Q <Q I>. Example 5: Let Z ={0,1,,,-1} be the rg of tegers modulo. <Z I> s the eutrosophc rg of modulo tegers. heorem.8:[9] Let <R I> be a eutrosophc rg. he <R I> s a rg. Note: We have sad teorem.. ad teorem.8. that a eutrosophc rg s a rg; but eutrosophc group may ot have a group structure. hs s a bg dfferece betwee these two algebrac structures. heorem.9:[] Let <R I> be a eutrosophc rg. <R I> s commutatve eutrosophc rg f a,b <R I>, ab=ba. Defto.10: [9] Let R I be a eutrosophc rg. proper subset K of R I s sad to be a eutrosophc subrg f K tself s a eutrosophc rg uder the operatos of R I. It s essetal that K = S I, a postve teger where S s a subrg of R..e. {P s geerated by the subrg S together wth I. ( Z + )}. Defto.11: [9] Let R I be a eutrosophc rg. We say R I s a eutrosophc rg of characterstc zero f x = 0 ( a teger) for all x R I s possble oly f = 0, the we call the eutrosophc rg to be a eutrosophc rg of characterstc zero. Example 6: Let Q I be the eutrosophc rg of ratoals. Q I s the eutrosophc rg of characterstc zero. Example 7: Cosder C I the eutrosophc rg of complex umbers. C I s the eutrosophc rg of characterstc zero. 3 Neutrosophc Modules Defto 3.1: Let (M,+,.) be ay R-module over a commutatve rg R ad let M(I)=<M I> be a eutrosophc set geerated by M ad I. he trple (M(I),+,.) s called a weak eutrosophc R-module over a rg R. If M(I) s a eutrosophc R-module over a eutrosophc rg R(I), the M(I) s called a strog eutrosophc R-module. he elemets of M(I) are called eutrosophc elemets ad the elemets of R(I) are called eutrosophc scalars. 184

186 Neutrosophc Operatoal Research Volume II ad If m = x+yi, m ı = z+ti M (I ) where x,y,z, t are elemets M ad α = u+vi R(I) where u, v are scalars R, we defe: m+m ı = (x+yi)+(z+ti) = (x+z)+(y+t)i, αm = (u+vi).(x+yi) = ux+(uy+xv+vy)i. Example 1: Let R be a commutatve rg, ad let J be a deal of R. (3.1) very mportat example of a eutrosphc R-module s R(I) tself: R(I) s, of course, a eutrosophc abela group, the multplcato R gves us a mappg. : RxR(I) > R(I), ad the eutrosphc rg axoms esure that ths scalar multplcato' turs R to a eutrosophc R-module. (3.) Sce J s closed uder addto ad uder multplcato by arbtrary elemets of R(I), t follows that J too s a eutrosophc R-module uder the addto ad multplcato of R. (3.3) R(I) s a weak eutrosophc R-module over a rg Q ad t s a strog eutrosophc R-module over a eutrosophc rg Q(I). (3.4) R (I) s a weak eutrosophc R-module over a rg R ad t s a strog eutrosophc R-module over a eutrosophc rg R(I). (3.5) M m (I) ={[a ]: a Q(I)} s a weak eutrosophc R-module over a rg Q ad t s a strog eutrosophc R-module over a eutrosophc rg Q(I) heorem 3.. Every strog eutrosophc R-module eutrosophc R-module. s a weak Proof: Suppose that M(I) s a strog eutrosophc module over a eutrosophc rg R(I). Sce R R (I) for every rg R, t follows that M(I) s a weak eutrosophc R-module. heorem 3.3. Every weak (strog) eutrosophc R-module s a R-module. Proof: Suppose that M(I) s a strog eutrosophc module over a eutrosophc rg R(I). Obvously, (M(I),+,.) s a abela group. Let m = x+yi, m ı = z + ti M(I ), α = a+bi, β = c+di R(I) where x,y,z,t M ad a, b, c, d R. he 185

187 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag (1) α (m + m ı ) = (a + bi )(x + yi + z + ti ) = ax + az + [ay + at + bx + by + bz + bt]i = (a+ bi)(x + yi) + (a+bi)(z + ti) =αm+αm ı. () (α + β )m = (a +bi + c+di)(x+yi) = ax + cx + [ay + cy + bx + dx + by + dy]i = (a + bi)(x +yi) + (c + di)(x +yi) =αm+βm (3) (αβ )m = ((a+ bi)(c+di))(x+yi) = acx +[acy + adx + bcx + bdx + ady +bcy + bdy]i = (a + bi)((c + di)(x + yi)) =α(βm) (4) For 1+1+0I R(I ), we have 1m = (1 + 0I)(x + yi ) = x(y )I = x+yi. ccordgly, M(I) s a R-module. Lemma 3.4. Let M(I) be a strog eutrosophc R-module over a eutrosophc rg R(I) ad let m=x+yi, m ı =z + ti, m ıı = u+vi M(I), α = a + bi R(I). he: (1) m+ m ıı = m ı + m ıı mples m= m ı. () α0=0. (3) 0m=0. (4) (-α)m=α(-m)=-(αm) Defto 3.5: Let M(I) be a strog eutrosophc R- module over a eutrosophc rg R(I) ad let N(I) be a oempty subset of M(I). N(I) s called a strog eutrosophc submodule of M(I) f N(I) s tself a strog eutrosophc R- module over R(I). It s essetal that N(I) cotas a proper subset whch s a R- module. 186

188 Neutrosophc Operatoal Research Volume II Defto 3.6: Let M(I) be a weak eutrosophc R-module over a rg R ad let N(I) be a oempty subset of M(I). N(I) s called a weak eutrosophc submodule of M(I), f N(I) s tself a weak eutrosophc R-module over R. It s essetal that N(I) cotas a proper subset whch s a R-module. heorem 3.7: Let M(I) be a strog eutrosophc R-module over a eutrosophc rg R(I) ad let N(I) be a oempty subset of M(I). N(I) s a strog eutrosophc submodule of M(I) f ad oly f the followg codtos hold: (1) m, m ı N(I) mples m + m ı N(I). () m N(I) mples αm N(I) for all α= a+bi R(I) a,b R. (3) N(I) cotas a proper subset whch s a R-module. Corollary 3.8 : Let M(I) be a strog eutrosophc R-module over a eutrosophc rg R(I) ad let N(I) be a oempty subset of M(I). N(I) s a strog eutrosophc submodule of M(I) f ad oly f the followg codtos hold: (1) m, m ı N(I) mples αm+βm ı N(I) for all α,β R(I). () N(I) cotas a proper subset whch s a R-module. Example. Let M(I) be a weak (strog) eutrosophc R-module. M(I) s a weak (strog) eutrosophc submodule called a trval weak (strog) eutrosophc submodule. Example 3. Let M(I) = R 3 (I)be a strog eutrosophc R-module over a eutrosophc rg R(I ) ad let N(I) ={(m=a+bi, m ı = c+di, 0 = 0+0I) M(I): a,b,c,d M}. he N(I) s a strog eutrosophc submodule of M(I). Example 4. Let M(I) = M m* (I) = {[ a ]: a R(I)} be a strog eutrosophc R- module over R(I) ad let N(I) = m* (I) ={[ b ]: b R(I) ad trace () = 0}. he N(I) s a strog eutrosophc submodule of M(I). heorem 3.9: Let M(I) be a strog eutrosophc R-module over a eutrosophc rg R(I) ad let {N (I)} be a famly of strog eutrosophc submodules of M(I). he N (I) s a strog eutrosophc submodule of M(I). 187

189 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Proof: Clearly 0 M N (I) ad N (I) Ø. Sce, for 0 M N (I Let bex,y N (I) ad let be ar-module. he x-y, ax N (I). Sce, for x-yn (Iad axn (IHece N (I) s a strog eutrosophc submodule of M(I). Remark 1. Let M(I) be a strog eutrosophc R-module over a eutrosophc rg R(I) ad let N 1 (I) ad N (I) be two dstct strog eutrosophc submodule of M(I). Geerally, N(I) N(I) s ot a strog eutrosophc submodule of M(I). However, f N 1 (I) N (I) or N (I) N 1 (I), the N 1 (I) N (I) s a strog eutrosophc submodule of M(I). Defto 3.10: Let M(I) ad N(I) be strog eutrosophc R-modules over a eutrosophc rg R(I) ad let :M (I )N(I) be a mappg of M(I) to N(I). s called a eutrosophc R-module homomorphsm f the followg codtos hold: (1) s a R-module homomorphsm. () (I ) I. If s a bectve eutrosophc R-module homomorphsm, the s called a eutrosophc R-module somorphsm ad we wrte M(I ) N(I). Defto 3.11: Let M(I) ad N(I) be strog eutrosophc R-modules over a eutrosophc rg R(I) ad let :M (I )N(I) be a eutrosophc R-module homomorphsm. 0}. (1) he kerel of deoted by Kers defed by the set {mm(i ) :(m) () he mage of deoted by Ims defed by the set {N(I) :(m) for some mm (I )}. Example 5. Let M(I) be a strog eutrosophc R-module over a eutrosophc rg R(I). (1) he mappg :M (I )M (I ) defed by (m) m for all mm (I ) s eutrosophc R-module homomorphsm ad Ker=0. () he mappg :M (I )M (I ) defed by (m) 0 for all mm (I ) s eutrosophc R-module homomorphsm sce I M (I ) but (I )

190 Neutrosophc Operatoal Research Volume II Defto 3.1: Let M(I) ad N(I) be strog eutrosophc R-modules over a eutrosophc rg R(I) ad let :M (I )N(I) be a eutrosophc R-module homomorphsm. he: of M. (1) Kers ot a strog eutrosophc submodule of M(I) but a submodule Proof. of M s clear. () Ims a strog eutrosophc submodule of N(I). () Clear. (1) Obvously, I M (I ) but (I ) 0. hat Kers a submodule Example 6. Let M be a module over the commutatve rg R. Suppose that N s a secod eutrosophc R-module, ad that f : M(I) N(I) s a homomorphsm of eutrosophc R-modules. he kerel of f, deoted by Ker f, s the set {mm : f(m) = 0 N }. Ker f s a submodule of M(I). Kerf = 0 f ad oly f f s a moomorphsm. heorem 3.13: Let N(I) be a strog eutrosophc submodule of a strog eutrosophc R-module M(I) over a eutrosophc rg R(I). Let :M(I)M(I)/N(I) be a mappg defed by (m) m N(I) for all mm (I ). he s ot a eutrosophc R-module homomorphsm. Proof. Obvous sce (I ) I N(I) N(I) I. heorem 3.14: Let N(I) be a strog eutrosophc submodule of a strog eutrosophc R-module M(I) over a eutrosophc rg R(I) ad let : M (I)K(I) be a eutrosophc R-module homomorphsm from M(I) to a strog eutrosophc R-module K(I) over R(I). If N(I) N(I) K(I) s the restrcto of to N(I) s defed by N(I) ) ) for all, the: (1) N(I) s a eutrosophc R-module homomorphsm. () N(I) Ker N(I) KerN(I). (3) Im N(I) (N(I)). Remark. If M(I) ad N(I) are strog eutrosophc R-modules over a eutrosophc rg R(I) ad,:m (I )N(I) are eutrosophc R-module homomorphsms, the () ad () are ot eutrosophc R-module homomorphsms sce ()(I ) (I ) (I ) I I I I ad 189

191 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag ()(I ) (I ) I I for all R(I ). Hece, f Hom(M (I ),N(I)) s the collecto of all eutrosophc R-module homomorphsms from M(I) to N(I), the Hom(M (I ),N(I)) s ot a eutrosophc R-module over R(I). hs s dfferet from what s obtaable the classcal R- module. Defto 3.15: Let K(I), M(I) ad N(I) be strog eutrosophc R-modules over a eutrosophc rg R(I) ad let :K(I )M(I ), :M(I )N(I) be eutrosophc R-module homomorphsms. he composto :K(I )N(I) s defed by (k) ((k)) for all kk(i ). heorem 3.16: Let K(I), M(I) ad N(I) be strog eutrosophc R-modules over a eutrosophc rg R(I) ad let :K(I )M (I ), :M (I )N(I) be eutrosophc R-module homomorphsms. he the composto :K(I )N(I) s a eutrosophc R-module homomorphsm. have: Proof: Clearly, s a R-module homomorphsm. For k I K(I ), we (I ) ((I )) (I ) I. Hece s a eutrosophc R-module homomorphsm. Corrolary 3.17: Let P(M(I)) be the collecto of all eutrosophc R- module homomorphsms from M(I) oto M(I). he () ()for all,, P(M (I )). heorem 3.18: Let K(I), M(I) ad N(I) be strog eutrosophc R-modules over a eutrosophc rg R(I) ad let :K(I )M (I ), :M (I )N(I) be eutrosophc R-module homomorphsms. he (1) If s ectve, the s ectve. () If s surectve, the s surectve. (3) If ad are ectve, the s ectve. 190

192 Neutrosophc Operatoal Research Volume II 4 Cocluso I ths paper we spred from the eutrosophc phlosophy whch F. Smaradache troduced the theory of eutrosophy Bascally we defed eutrosophc R-modules ad eutrosophc submodules whch are completely dfferet from the classcal module ad submodule the structural propertes. It was show that every weak eutrosophc R-module s a R-module ad every strog eutrosophc R-module s a R-module. Fally, eutrosophc quotet modules ad eutrosophc R-module homomorphsm are explaed ad some deftos ad theorems are gve. Refereces [1] gboola, kwu D ad Oyebo Y (01) Neutrosophc Groups ad Neutrosophc Subgroups, It. J Math. Comb [] gboola, kwu D ad Oyebo Y (011) Neutrosophc Rgs I, It. J. Math. Comb [3] gboola, deleke EO ad kleye S (01) Neutrosophc Rgs II, It. J. Math. Comb 1-8. [4] gboola ad kleye S (014) Neutrosophc Vector Space Neutrosophc Sets ad Systems, Vol [5] Yetk E, Olgu N. (014) New ype Fuzzy Module over Fuzzy Rgs, he Scetfc World Joural, DOI: /014/73093 [6] Smaradache F, (005) Ufyg Feld Logcs. Neutrosophy, Neutrosophc Probablty, Set ad Logc. Rehoboth: merca Research Press [7] Uluçay V, Şah M, Olgu N, Klcma (017) O eutrosophc soft lattces, frca Matematka, Jue 017, Volume 8, Issue 3 4, pp [8] Sharp RY, (000). Steps Commutatve lgebra, (Secod edto), Cambrdge Uversty Press [9] Smaradache, F. l, M., (016) Neutrosophc rplet Group, Neural Comput & pplc do: /s x [10] Kadasamy WBV, Smaradache F, (006) Neutrosophc Rgs, Hexs, Phoex, rzoa [11] Kadasamy WBV, Smaradache F, (004) Basc Neutrosophc lgebrac Structures ad ther pplcatos to Fuzzy ad Neutrosophc Models, Hexs, Phoex, rzoa [1] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [13] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [14] bdel-basset, Mohamed, Ma Mohamed, ad ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." Joural of mbet Itellgece ad Humazed Computg (017): DOI: [15] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [16] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers:

193 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [17] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [18] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [19] bdel-baset, Mohamed, Ibrahm M. Hezam, ad Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [0] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [1] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016): [] Mumtaz l, Floret Smaradache, Muhammad Shabr: Soft Neutrosophc Groupods ad her Geeralzato, Neutrosophc Sets ad Systems, vol. 6, 014, pp do.org/10.581/zeodo [3] Muhammad Shabr, Mumtaz l, Muazza Naz, Floret Smaradache: Soft Neutrosophc Group, Neutrosophc Sets ad Systems, vol. 1, 013, pp do.org/10.581/zeodo [4] Mumtaz l, Floret Smaradache, Luge Vladareau, Muhammad Shabr: Geeralzato of Soft Neutrosophc Rgs ad Soft Neutrosophc Felds, Neutrosophc Sets ad Systems, vol. 6, 014, pp do.org/10.581/zeodo [5] Mumtaz l, Floret Smaradache, Luge Vladareau, Muhammad Shabr: Geeralzato of Soft Neutrosophc Rgs ad Soft Neutrosophc Felds, Neutrosophc Sets ad Systems, vol. 6, 014, pp do.org/10.581/zeodo

194 Neutrosophc Operatoal Research Volume II X Neutrosophc rplet Ier Product Mehmet Şah 1 bdullah Kargı 1 Departmet of Mathematcs, Gazatep Uversty, Gazatep, 7310, urkey. Emal: mesah@gatep.edu.tr Departmet of Mathematcs, Gazatep Uversty, Gazatep, 7310, urkey. Emal: abdullahkarg7@gmal.com bstract I ths paper, the oto of eutrosophc trplet er product s gve ad propertes of eutrosophc trplet er product spaces are studed. Furthermore, we show that ths eutrosophc trplet oto s dfferet from the classcal oto. Keywords Neutrosophc trplet er product; Neutrosophc trplet metrc spaces; Neutrosophc trplet vector spaces; Neutrosophc trplet ormed spaces. 1 Itroducto he cocept of eutrosophc logc ad eutrosophc set were troduced by Smaradache [1]. I ths cocept, sets have truth fucto, falsty fucto ad determacy fucto. hese fuctos defed as depedet o each other. herefore, the cocept overcomes may ucertates our daly lfe. I fact, Zadeh troduced the cocept of fuzzy set [] ad taassov troduced the cocept of tutostc fuzzy set [3] to overcome ucertates. he fuzzy set has oly truth (membershp) fucto. he tutostc fuzzy set has truth fucto, falsty fucto ad determacy fucto. But these fuctos defed as depedet o each other. herefore, eutrosophc set s the geeralzato of fuzzy set ad tutostc fuzzy set. Smaradache at al. troduced eutrosophc algebrac structures [4, 5] usg the eutrosophc theory; Smaradache at al. troduced eutrosophc trplet theory ad eutrosophc trplet groups [6-8]. he eutrosophc trplet set s completely dfferet from the classcal sets, sce for each elemet a eutrosophc trplet set N together wth a bary operato *; there exst a eutral of a called eut(a) where a*eut(a)=eut(a)*a=a ad a opposte of a called at(a) where 193

195 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag a*at(a)=at(a)*a=eut(a). he eut(a) s dfferet from the classcal algebrac utary elemet. eutrosophc trplet s of the form <a, eut(a), at(a)>. lso; Smaradache at al. studed the eutrosophc trplet rg [9] ad the eutrosophc trplet feld [10]. Şah at al. studed eutrosophc trplet metrc space, eutrosophc trplet vector space ad eutrosophc trplet ormed space [11]. Recetly some researchers have bee dealg wth eutrosophc set theory. For example; Smaradache at al. studed the sgle valued eutrosophc graphs [1,30-39], terval valued eutrosophc graphs [13] ad SV- rapezodal eutrosophc umbers [14]. Lu at al. studed terval eutrosophc hestat set [15], eutrosophc ucerta lgustc umber [16], some power geeralzed aggregato operators based o the terval eutrosophc umbers [17], mult-crtera group decso-makg based o terval eutrosophc ucerta lgustc varables [18], terval eutrosophc prortzed OW operator ad aggregato operators based o rchmedea t-coorm ad t-orm for the sgle valued eutrosophc umbers [19], multple attrbute decso makg method based o ormal eutrosophc geeralzed weghted power averagg operator [0] ad mult-valued eutrosophc umber boferro mea operators [1]. lso, [-9] eutrosophc set theory was studed. I ths paper, we troduced eutrosophc trplet er product space. lso, we gve ew propertes ad ew deftos for ths structure. I ths paper, secto, some prelmary results for eutrosophc trplet sets, eutrosophc trplet rg ad feld, eutrosophc trplet metrc space, eutrosophc trplet vector space ad eutrosophc trplet ormed space are gve. I secto 3, eutrosophc trplet er product space s defed ad some propertes of a eutrosophc trplet er product space are gve. It s show that eutrosophc trplet er product dfferet from the classcal er product. lso, t s show that f certa codtos are met; every eutrosophc trplet er product space ca be a eutrosophc trplet ormed space ad eutrosophc trplet metrc space at the same tme. Furthermore, the covergece of a sequece ad a Cauchy sequece a eutrosophc trplet er product space are defed. I secto 4, coclusos are gve. Prelmares Defto.1: (Smaradache at al. [016]) Let N be a set together wth a bary operato *. he, N s called a eutrosophc trplet set f for ay a N, there exsts a eutral of a called eut(a), dfferet from the classcal algebrac utary elemet, ad a opposte of a called at(a), wth eut(a) ad at(a) belogg to N, such that: a*eut(a)= eut(a)* a=a, 194

196 Neutrosophc Operatoal Research Volume II ad a*at(a)= at(a)* a=a. he elemets a, eut(a) ad at(a) are collectvely called as eutrosophc trplet, ad we deote t by (a, eut(a), at(a)). Here, we mea eutral of a ad apparetly, a s ust the frst coordate of a eutrosophc trplet ad t s ot a eutrosophc trplet. For the same elemet a N, there may be more eutrals to t eut(a) ad more oppostes of t at(a). Defto. (Smaradache at al. [016]): Let (N,*) be a eutrosophc trplet set. he, N s called a eutrosophc trplet group, f the followg codtos are satsfed. 1) If (N,*) s well-defed,.e. for ay a, b N, oe has a*b N. ) If (N,*) s assocatve,.e. (a*b)*c= a*(b*c) for all a, b, c N. he eutrosophc trplet group, geeral, s ot a group the classcal algebrac way. Oe ca cosder that eutrosophc eutrals are replacg the classcal utary elemet, ad the eutrosophc oppostes are replacg the classcal verse elemets. Defto.3: (Smaradache at al. [016]) Let (N,*) be a eutrosophc trplet group. he N s called a commutatve eutrosophc trplet group f for all a, b N, we have a*b=b*a. Proposto.4: (Smaradache at al. [016]) Let (N,*) be a eutrosophc trplet group wth respect to * ad a,b,c N; 1) a*b= a*c f ad oly f eut(a)*b=eut(a)*c ) b*a= c*a f ad oly f b*eut(a)=c*eut(a) 3) f at(a)*b=at(a)*c, the eut(a)*b=eut(a)*c 4) f b*at(a)=c*at(a), the b*eut(a)=c*eut(a) heorem.5: (Smaradache at al. [016]) Let (N,*) be a commutatve eutrosophc trplet group wth respect to * ad a, b N; ) eut(a)*eut(b)= eut(a*b); ) at(a)*at(b)= at(a*b); heorem.6: (Smaradache at al. [016]) Let (N,*) be a commutatve eutrosophc trplet group wth respect to * ad a N; ) eut(a)*eut(a)= eut(a); ) at(a)*eut(a)=eut(a)* at(a)= at(a); 195

197 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Defto.7: (Smaradache at al. [017]) Let (NF,*,#) be a eutrosophc trplet set together wth two bary operatos * ad #. he (NF,*, #) s called eutrosophc trplet feld f the followg codtos hold. 1. (NF,*) s a commutatve eutrosophc trplet group wth respect to *.. (NF, #) s a eutrosophc trplet group wth respect to #. 3. a#(b*c)= (a#b)*(a#c) ad (b*c)#a = (b#a)*(c#a) for all a,b,c NF. heorem.8: (Şah at al. [017]) Let (N,*) be a eutrosophc trplet group wth o zero dvsors ad wth respect to *. For a N; If a = eut(a), the there exsts a at(a) such that eut(a) = at(a) = a. heorem.9: (Şah at al. [017]) Let (N,*) be a eutrosophc trplet group wth o zero dvsors ad wth respect to *. For a N; ) eut(eut(a))= eut(a) ) at(eut(a))= eut(a)) ) at(at(a))= a v) eut(at(a))= eut(a) Defto.10: (Şah at al. [017]) Let (N,*) be a eutrosophc trplet set ad let x*y N for all x, y N. If the fucto d: NxN R + {0} satsfes the followg codtos; d s called a eutrosophc trplet metrc. For all x, y, z N; a) d(x, y) 0; b) If x=y; the d(x, y)=0 c) d(x, y)= d(y, x) d) If there exsts ay elemet y N such that d(x, z) d(x, z*eut(y)), the d(x, z*eut(y)) d(x, y)+ d(y, z). Furthermore; ((N,*), d) space s called eutrosophc trplet metrc space. Defto.11:(Şah at al. [017]) Let (NF, 1, # 1 ) be a eutrosophc trplet feld ad let (NV,, # ) be a eutrosophc trplet set together wth bary operatos " ad #. he (NV,, # ) s called a eutrosophc trplet vector space f the followg codtos hold. For all u, v NV ad for all k NF; such that u v NV ad u # k NV ; 1) (u v) t= u (v t), for every u, v, t NV ) u v = v u, for every u, v NV 3) (v u) # k= (v# k) (u# k), for all k NF ad for all u, v NV 196

198 Neutrosophc Operatoal Research Volume II 4) (k 1 t) # u= (k# v) 1 (u# v), for all k,t NF ad for all u NV 5) (k# 1 t) # u= k# 1 (t# u), for all k,t NF ad for all u NV 6) For all u NV; such that u # eut(k)= eut(k) # u = u, there exsts ay eut(k) NF Here; the codto 1) ad ) dcate that the eutrosophc trplet set (NV, ) s a commutatve eutrosophc trplet group. Example.1: (Şah at al. [017]) Let X={1,} be a set ad P(X)={, {1}, {}, {1, }} be power set of X ad let (P(X), *) be a eutrosophc trplet set. Where *=, eut( )= eut({1})= eut({}) =, eut({1, })= {1} ad at()= for P(X) ad for *. he (P(X),, ) s a eutrosophc trplet feld, sce for eut()=, at()= for,. Now, we show that (P(X), *, ) s a eutrosophc trplet vector space o (P(X),, ) eutrosophc trplet feld. Defto.13: (Şah at al. [017]) Let (NV,, # ) be a eutrosophc trplet vector space o (NF, 1, # 1 ) eutrosophc trplet feld. If. :NV R + {0} fucto satsfes followg codto;. s called eutrosophc trplet ormed o (NV,, # ). Where; f: NF X NV R + {0}, f(α,x)= f(at(α), at(x)) s a fucto ad for every x, y NV ad α NF; NV. a) x 0; b) If x=eut(x), the x =0 c) α# x = f(α,x). x d) at(x) = x e) If x y x y eut(k) ; the x y eut(k) x + y, for ay k Furthermore o (NV,, # ), the eutrosophc trplet vector space defed by. s called a eutrosophc trplet ormed space ad s deoted by ((NV,, # ),. ). Proposto.14: (Şah et al. [017])Let ((NV,, # ),. ) be a eutrosophc trplet ormed space o (NF, 1, # 1 )eutrosophc trplet feld. he, the fucto d: NV x NV R defed by d(x, y) = x at(y) provdes eutrosophc trplet metrc space codtos. 197

199 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag 3 Neutrosophc rplet Ier Product Space Now let s defe the eutrosophc trplet er product spaces o the eutrosophc trplet vector space. Defto 3.1: Let (NV,, # ) be a eutrosophc trplet vector space o (NF, 1, # 1 ) eutrosophc trplet feld. If <.,.> :NV x NV R + {0} fucto satsfes followg codto; <.,.> s called eutrosophc trplet er product o (NV,, # ). Where; f: NF X NV X NV R + {0}, f(α, x, y)= f(at(α), at(x), at(y)) ad f(α, x, y)= f(α,y, x), s a fucto ad for every x, y NV ad α, β NF; a) <x, x> 0; b) If x=eut(x), the <x, x> =0 c) <(α# x) (β# y), z> = f(α, x, z). <x, z>+ f(β,y,z). <y, z> d) <at(x), at(x)>= <x, x> e) <x, y>=<y, x> Furthermore o (NV,, # ), the eutrosophc trplet vector space defed by <.,.> s called a eutrosophc trplet er product space ad s deoted by ((NV,, # ), <.,.>). Corollary 3.: It s clear by defto 3.1 that eutrosophc trplet er product spaces are geerally dfferet from classcal er product spaces, sce for there s ot ay f fucto classcal er product space. Example 3.3: From example.1; (P(X), *, ) s a eutrosophc trplet vector space o (P(X),, ) eutrosophc trplet feld. he takg f: P(X) X P(X) X P(X) R + {0}, f(,b,c)= s(( B)\C)/s(B\C),. :P(X) R + {0}. Now we show that, <, B>= s((\b) (B\)) s a eutrosophc trplet er product ad ((P(X),*, ),. ) s a eutrosophc trplet ormed space. Where; s() s umber of elemets P(X)ad eut()=, at()= for * ad *B= B a) <, B>= s( B) 0. b) If =eut()=, the <, >= 0. c) It s clear that <( B ) (C D), E> = f(, B, E). <B, E> + f(c, D, E). <D, E> d) s, = at(), t s clear that <at(), at()>= <, >. 198

200 Neutrosophc Operatoal Research Volume II e) It s clear that <, B>= s((\b) (B\))= s((b\) (\B))= <B, >. heorem 3.4: Let (NV,, # ) be a eutrosophc trplet vector space o (NF, 1, # 1 ) eutrosophc trplet feld ad let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NV,, # ) ad f: NF X NV X NV R + {0}, f(α,x,y)= f(at(α), at(x), at(y)) s a fucto ad for every x, y NV ad α, β NF; he; <(α# x) (β# y), (α# x) (β# y)>= f(α, (α# x) (β# y), x,).f(α, x, x,).<x, x>+ [ f(α, (α# x) (β# y), x,). f(β, x, y,)+ f(β, (α# x) (β# y), y).f(α, x, y,)].<x, y>+ f(β, (α# x) (β# y), y). f(β, y, y,).<y, y> Proof. <(α# x) (β# y), (α# x) (β# y)>= f(α,x, (α# x) (β# y)).<x, (α# x) (β# y) >+ f(β,y, (α# x) (β# y)).<y, (α# x) (β# y) >. From the defto 3.1; sce <x, y>= <x, y> ad f(α,x,y)= f(α,y, x); f(α,x, (α# x) (β# y)).<x, (α# x) (β# y) >+ f(β,y, (α# x) (β# y)).<y, (α# x) (β# y) >= f(α, (α# x) (β# y), x,).< (α# x) (β# y), x >+ f(β, (α# x) (β# y), y).< (α# x) (β# y), y > = f(α, (α# x) (β# y), x,).[ f(α, x, x,).<x, x>+ f(β, x, y,).<x, y>] + f(β, (α# x) (β# y), y).[ f(α, x, y,).<x, y>+ f(β, y, y,).<y, y>] = f(α, (α# x) (β# y), x).f(α, x, x,).<x, x>+ [ f(α, (α# x) (β# y), x,). f(β, x, y,)+ f(β, (α# x) (β# y), y).f(α, x, y,)].<x, y>+ f(β, (α# x) (β# y), y). f(β, y, y,).<y, y> heorem 3.5: Let (NV,, # ) be a eutrosophc trplet vector space o (NF, 1, # 1 ) eutrosophc trplet feld ad let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NV,, # ) ad f: NF X NV X NV R + {0}, f(α,x,y)= f(at(α), at(x), at(y)) s a fucto ad for every x,y NV ad α NF. If eut(x)= eut(y) the; (< x, y >) <x, x>.<y, y> 199

201 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Proof: It s clear that f x = eut(x) or y = eut(y) the; (< x, y >) <x, x>.<y, y>. We suppose that x eut(x). From the theorem 3.4; f f(α, (α# x) (β# y), x,) = f(α, x, x)= f(α, x, x)= <x,y> <x,x>, f(β, (α# x) (β# y), y)= f(β, y, y,)= f(β, x, y,) =1 are take; 0 <(α# x) (β# y), (α# x) (β# y) > = f(α, (α# x) (β# y), x).f(α, x, x,).<x, x>+ [ f(α, (α# x) (β# y), x,). f(β, x, y,)+ f(β, (α# x) (β# y), y).f(α, x, y,)].<x, y>+ f(β, (α# x) (β# y), y). f(β, y, y,).<y, y>= ( <x,y> <x,x> ). <x, x>- <x,y>.<x,y> <x,x> (<x,y>) <x,x> 0 <y, y> - (<x,y>) <x,x> - (<x,y>) <x,x> ad - (<x,y>) <x,x> 0 < x, x >.<y, y> - < x, x >. (<x,y>) <x,x> (< x, y >) < x, x >.<y, y>. - <x,y>.<x,y> + <y, y>= <x,x> (<x,y>) +<y, y>= <y, y> -. hus; we have <x,x> heorem 3.6: Let (NV,, # ) be a eutrosophc trplet vector space o (NF, 1, # 1 ) eutrosophc trplet feld ad let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NV,, # ) ad f: NF X NV X NV R + {0}, f(α,x,y)= f(at(α), at(x), at(y)) s a fucto ad for every x, y NV ad α NF. If f(α,x,x)= f(α,x) ad x = < x, x > 1. he; ((NV,, # ),. ) s a eutrosophc trplet ormed space o (NV,, # ). Proof: s ((NV,, # ), <.,.>) s a eutrosophc trplet er product space ad f(α,x,x)= f(α,x), we have; a) x = < x, x > 1 0. b) If x= eut(x) the; < x, x > 1 = x = 0. c) α# x = < α# x, α# x > 1 = f(α, x, x) 1. f(α, x, x) 1. < x, x > 1 = f(α, x, x). < x, x > 1 = f(α, x). x d) at(x) = < at(x), at(x) > 1 =< x, x > 1 = x e) From the theorem 3.4; f f(α, (α# x) (β# y), x,) = f(α, x, x)= f(α, x, x)= f(β, (α# 00

202 Neutrosophc Operatoal Research Volume II x) (β# y), y)= f(β, y, y,)= f(β, x, y,) =1 ad <(α# x) (β# y), (α# x) (β# y)> = <x y, x y > are take; x# y = <(α# x) (β# y), (α# x) (β# y)> = <x y, x y > = f(α, (α# x) (β# y), x).f(α, x, x,).<x, x>+ [ f(α, (α# x) (β# y), x,). f(β, x, y,)+ f(β, (α# x) (β# y), y).f(α, x, y,)].<x, y>+f(β, (α# x) (β# y), y). f(β, y, y,).<y, y>= <x,x> +.<x,y>+<y,y>= x +.<x,y>+ y. From the theorem 3.5; f eut(x)= eut(y) the; (< x, y >) <x, x>.<y, y> x +.<x,y>+ y x + x y + y = ( x + y ). Sce eut(x)= eut(y); t s clear that x y x y eut(k). Where we ca take eut(k)= eut(x).hus; x y eut(k) x + y Corollary 3.7: Let ((NV,, # ),. ) be a eutrosophc trplet ormed space o (NF, 1, # 1 )eutrosophc trplet feld ad let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NF, 1, # 1 )eutrosophc trplet feld such that x = < x, x > 1. he, the fucto d: NV x NV R defed by d(x, y) = x at(y) = < x at(y), x at(y) > 1 provdes eutrosophc trplet metrc space codtos. Proof: It s clear that from proposto.14. Corollary 3.8: Every eutrosophc trplet metrc space s reduced by a eutrosophc trplet er product space. But the opposte s ot always true. Smlarly; Every eutrosophc trplet ormed space s reduced by a eutrosophc trplet er product space. But the opposte s ot always true. Defto 3.9: Let ((NV,, # ),. ) be a ((NV,, # ), <.,.>) ormed space o (NF, 1, # 1 )eutrosophc trplet feld ad ((NV,, # ), <.,.>) be a eutrosophc trplet er product space such that x = < x, x > 1. d: NVx NV R eutrosophc trplet metrc defe by d(x, y)= x at(y) = < x at(y), x at(y) > 1 s called the eutrosophc trplet er product space reduced by (NV,, # ). Now let s defe the covergece of a sequece ad a Cauchy sequece the eutrosophc trplet er space wth respect to eutrosophc trplet metrc whch s reduced by eutrosophc trplet er space. 01

203 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag Defto 3.10: Let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NF, 1, # 1 ) eutrosophc trplet feld, {x } be a sequece ths space ad d be a eutrosophc trplet metrc reduced by ((NV,, # ), <.,.>). For all ε>0, x NV such that for all M d(x, {x })= < x at({x }), x at({x }) > 1 < ε f there exsts a M N; {x } sequece coverges to x. It s deoted by lm x = x or x x Defto 3.11: Let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NF, 1, # 1 ) eutrosophc trplet feld, {x } be a sequece ths space ad d be a eutrosophc trplet metrc reduced by ((NV,, # ), <.,.>). For all ε>0, x NV such that for all M d(({x m }, {x })= x at({x } ) < ({x m } at({x }), ({x m } at({x }) > 1 < ε f there exsts a M N; {x } sequece s called Cauchy sequece. Defto 3.1: Let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NF, 1, # 1 ) eutrosophc trplet feld, {x } be a sequece ths space ad d be a eutrosophc trplet metrc reduced by ((NV,, # ), <.,.>). If each {x } cauchy sequece ths space s coverget to d reduced eutrosophc trplet metrc; ((NV,, # ), <.,.>) s called eutrosophc trplet Hlbert space. heorem 3.13: Let ((NV,, # ), <.,.>) be a eutrosophc trplet er product space o (NF, 1, # 1 ) eutrosophc trplet feld, ad {x } ad {y } be sequeces ((NV,, # ), <.,.>) such that {x } x NV ad {y } y NV, the; lm < x, y >= <x,y> Proof: < x, y > < x, y > = < x, y > < x, y > +< x, y > < x, y > < x, y > < x, y > + < x, y > < x, y > = < x, y y > + < x x, y >. From theorem 3.5; as (< x, y >) <x, x>.<y, y> ad from defto 3.9; as d(x, y)= x at(y) = < x at(y), x at(y) > 1, < x, y y > + < x x, y > x y y + x x y. s {x } x ad {y } y; lm < x, y >= <x, y> 0

204 Neutrosophc Operatoal Research Volume II 4 Cocluso I ths paper, we troduced eutrosophc trplet er product space. We also show that ths eutrosophc trplet oto dfferet from the classcal oto. hs eutrosophc trplet oto has several extraordary propertes compared to the classcal oto. We also studed some terestg propertes of ths ewly bor structure. We gve rse to a ew feld or research called eutrosophc trplet er product space. Refereces [1] Zadeh Lotf., Fuzzy sets." Iformato ad cotrol, (1965), pp , [] taassov,. K., (1986) Itutostc fuzzy sets, Fuzzy Sets Syst, (1986) 0:87 96 [3] Smaradache F., Ufyg Feld logcs, Neutrosophy: Neutrosophc Probablty, Set ad Logc, merca Research Press (1999) [4] Kadasamy WBV, Smaradache F., Basc eutrosophc algebrac structures ad ther applcatos to fuzzy ad eutrosophc models, Hexs, Frotga (004) p 19 [5] Kadasamy WBV., Smaradache F., Some eutrosophc algebrac structures ad eutrosophc -algebrac structures. Hexs, Frotga (006) p 19 [6] Smaradache F., & l M., Neutrosophc trplet as exteso of matter plasma, umatter plasma ad atmatter plasma, PS Gaseous Electrocs Coferece (016), do: /BPS.016.GEC.H6.110 [7] Smaradache F., & l M., he Neutrosophc rplet Group ad ts pplcato to Physcs, preseted by F. S. to Uversdad Nacoal de Qulmes, Departmet of Scece ad echology, Beral, Bueos res, rgeta (0 Jue 014) [8] Smaradache F., & l M., Neutrosophc trplet group. Neural Computg ad pplcatos, (016) 1-7. [9] Smaradache F., & l M., Neutrosophc rplet Feld Used Physcal pplcatos, (Log Number: NWS ), 18th ual Meetg of the PS Northwest Secto, Pacfc Uversty, Forest Grove, OR, US (Jue 1-3, 017) [10] Smaradache F., & l M., Neutrosophc rplet Rg ad ts pplcatos, (Log Number: NWS ), 18th ual Meetg of the PS Northwest Secto, Pacfc Uversty, Forest Grove, OR, US (Jue 1-3, 017). [11] Şah, M., ad Kargı,., Neutrosophc trplet ormed space, Ope Physcs, (017). ccepted [1] Broum S., Bakal., alea M., Smaradache F., Sgle Valued Neutrosophc Graphs: Degree, Order ad Sze. IEEE Iteratoal Coferece o Fuzzy Systems (FUZZ) (016), pp [13] Broum S., Bakal., alea M., Smaradache F. Decso-Makg Method Based O the Iterval Valued Neutrosophc Graph, Future echologe, IEEE (016), pp [14] Broum S., Bakal., alea M., Smaradache F., Vladareau L. Computato of Shortest Path Problem a Network wth SV-rapezodal Neutrosophc Numbers, Proceedgs of the 016 Iteratoal Coferece o dvaced Mechatroc Systems, Melboure, ustrala, (016) pp [15] Lu P. ad Sh L., he Geeralzed Hybrd Weghted verage Operator Based o Iterval Neutrosophc Hestat Set ad Its pplcato to Multple ttrbute Decso Makg, Neural Computg ad pplcatos (015) 6():

205 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag [16] Lu P. ad Sh L., Some Neutrosophc Ucerta Lgustc Number Heroa Mea Operators ad her pplcato to Mult-attrbute Group Decso makg, Neural Computg ad pplcatos (015) do: /s [17] Lu P., ag G., Some power geeralzed aggregato operators based o the terval eutrosophc umbers ad ther applcato to decso makg, Joural of Itellget & Fuzzy Systems 30 (016) [18] Lu P. ad ag G., Mult-crtera group decso-makg based o terval eutrosophc ucerta lgustc varables ad Choquet tegral, Cogtve Computato, (016) 8(6) [19] Lu P. ad Wag Y., Iterval eutrosophc prortzed OW operator ad ts applcato to multple attrbute decso makg, oural of systems scece & complexty (016) 9(3): [0] Lu P., eg F., Multple attrbute decso makg method based o ormal eutrosophc geeralzed weghted power averagg operator, teral oural of mache learg ad cyberetcs (015) /s y [1] Lu P., Zhag L., Lu X., ad Wag P., Mult-valued Neutrosophc Number Boferro mea Operators ad her pplcato Multple ttrbute Group Decso Makg, teral oural of formato techology & decso makg (016) 15(5) [] Sah M., ad Kargı., Neutrosophc trplet metrc space ad eutrosophc trplet ormed space, ICMME -017, Şalıurfa. [3] Sah M., Del I., I, ad Ulucay V., Jaccard vector smlarty measure of bpolar eutrosophc set based o mult-crtera decso makg. I: Iteratoal coferece o atural scece ad egeerg (ICNSE 16), (016) March 19 0, Kls [4] Sah M., Del I., ad Ulucay V., Smlarty measure of bpolar eutrosophc sets ad ther applcato to multple crtera decso makg, Neural Comput & pplc (016) DOI /S0051 [5] Lu C. ad Luo Y., Power aggregato operators of smplfeld eutrosophc sets ad ther use mult-attrbute group decso makg, İEE/C Joural of utomatca Sca (017) Vol pp, ssue: 99, DOI: /JS [6] Sah R. ad Lu P., Some approaches to mult crtera decso makg based o expoetal operatos of smpled eutrosophc umbers, Joural of Itellget & Fuzzy Systems (017) vol. 3, o. 3, pp , DOI: /JIFS [7] Lu P. ad L H., Mult attrbute decso-makg method based o some ormal eutrosophc boferro mea operators, Neural Computg ad pplcatos (017) vol. 8, ssue 1, pp , DOI /s z [8] Şah M., Olgu N., Uluçay V., Kargı., ad Smaradache, F., ew smlerty measure o falsty value betwee sgle valued eutrosophc sets based o the cetrod pots of trasformed sgle valued eutrosophc umbers wth applcatos to patter recogto, Neutrosophc Sets ad Systems, (017) vol 15. Pp 31-48, do: org/10.581/zeodo [9] Şah M., Ecemş O., Uluçay V., ad Kargı,., Some ew geeralzed aggregato operators based o cetrod sgle valued tragular eutrosophc umbers ad ther applcatos mult-attrbute decso makg, sa Joural of Mathematcs ad Computer Research (017) 16(): [30] bdel-basset, Mohamed, et al. " ovel group decso-makg model based o tragular eutrosophc umbers." Soft Computg (017): DOI: [31] Hussa, bdel-nasser, et al. "Neutrosophc Lear Programmg Problems." Peer Revewers: 15. [3] bdel-basset, Mohamed, Ma Mohamed, ru Kumar Sagaah. "Neutrosophc HP-Delph Group decso makg model based o trapezodal eutrosophc umbers." 04

206 Neutrosophc Operatoal Research Volume II Joural of mbet Itellgece ad Humazed Computg (017): DOI: [33] Mohamed, Ma, et al. " Crtcal Path Problem Neutrosophc Evromet." Peer Revewers: 167. [34] Mohamed, Ma, et al. " Crtcal Path Problem Usg ragular Neutrosophc Number." Peer Revewers: 155. [35] Mohamed, Ma, et al. "Usg Neutrosophc Sets to Obta PER hree-mes Estmates Proect Maagemet." Peer Revewers: 143. [36] Mohamed, Ma, et al. "Neutrosophc Iteger Programmg Problem." Neutrosophc Sets & Systems 15 (017). [37] bdel-baset, Mohamed, Ibrahm M. Hezam, Floret Smaradache. "Neutrosophc goal programmg." Neutrosophc Sets Syst 11 (016): [38] Hezam, Ibrahm M., Mohamed bdel-baset, ad Floret Smaradache. "aylor seres approxmato to solve eutrosophc multobectve programmg problem." Neutrosophc Sets ad Systems 10 (015): [39] El-Hefeawy, Nacy, et al. " revew o the applcatos of eutrosophc sets." Joural of Computatoal ad heoretcal Naoscece 13.1 (016): [40] I. rokara, R. Dhavaseela, S. Jafar, M. Parmala: O Some New Notos ad Fuctos Neutrosophc opologcal Spaces, Neutrosophc Sets ad Systems, Vol. 16 (017), pp do.org/10.581/zeodo [41] R. Dhavaseela, S. Jafar, R. Narmada Dev, Md. Haf Page: Neutrosophc Bare Spaces, Neutrosophc Sets ad Systems, Vol. 16 (017), pp do.org/10.581/zeodo [4] P. Iswarya, K. Bageerath: Study o Neutrosophc Froter ad Neutrosophc Sem-froter Neutrosophc opologcal Spaces, Neutrosophc Sets ad Systems, Vol. 16 (017), pp do.org/10.581/zeodo [43]... gboola, S.. kleye: Neutrosophc Vector Spaces, Neutrosophc Sets ad Systems, vol. 4, 014, pp do.org/10.581/zeodo

207 Edtors: Prof. Floret Smaradache Dr. Mohamed bdel-basset Dr. Vctor Chag hs book treats all kd of data eutrosophc evromet, wth real-lfe applcatos, approachg topcs as lear programmg problem, lear fractoal programmg, teger programmg, tragular eutrosophc umbers, sgle valued tragular eutrosophc umber, eutrosophc optmzato, goal programmg problem, aylor seres, mult-obectve programmg problem, eutrosophc geometrc programmg, eutrosophc topology, eutrosophc ope set, eutrosophc sem-ope set, eutrosophc cotuous fucto, cyldrcal sk plate desg, eutrosophc MULIMOOR, alteratve solutos, decso matrx, rato system, referece pot method, full multplcatve form, ordal domace, stadard error, market research, ad so o. he selected papers deal wth the allevato of world chages, cludg chagg demographcs, acceleratg globalzato, rsg evrometal cocers, evolvg socetal relatoshps, growg ethcal ad goverace cocer, expadg the mpact of techology; some of these chages have mpacted egatvely the ecoomc growth of prvate frms, govermets, commutes, ad the whole socety. 06