Neutrosophic linear goal progamming by Mia Amalia

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Global JJournal of f EngineeringScience and RessearchManagement a t G PROGRAMMIN NG NEUTROSOPHIC LINEAR GOAL Surapati Pramanik* *Departm ment of Mathematics, Nanndalal Ghoshh B.T. Colleg ge, Panpur, P.O.-Narayanp P pur, District – North 24 Parganas, P Pinn code-743126, West Benggal, India DOI: KEYWOR RDS:Goal prrogramming, fuzzy goal programming, p intuitionisticc fuzzy goall programminng, neutrosophhic goal prograamming, neutroosophic set, sinngle valued neu utrosophic set.

ACT ABSTRA This paperr proposes the framework f of neutrosophic linear l goal prog gramming (NG GP) approach for f solving muulti objective optimization o prroblems involvving uncertaintty and indeterm minacy. In the pproposed apprroach, the degrree of membeership (acceptaance), indeterrminacy and falsity (rejectiion) of the oobjectives are simultaneoussly consideredd. Three neutroosophic linear goal program mming models have been prooposed. The drrawbacks of thhe existing neeutrosophic opptimization models have beenn addressed annd new directioon of research in neutrosophhic optimizatio on problem haas been proposeed. The essencce of the propoosed approach is that it is caapable of dealinng with indeteerminacy and falsity f simultanneously.

INTROD DUCTION Goal progrramming can be b viewed in tw wo ways. In firsst consideration n, it is an extennsion of linear programming to include muulti objectives, expressed by means of attem mpted achievem ment of goal vaalues. In seconnd consideratioon, linear prog gramming is a special case of o goal program mming having single objectiive. These two considerationns reflect thatt goal program mming lies withhin the paradiggm of multi objjective program mming [1]. Gooal programminng may be chharacterized as an analytical approach deviised to address multi objectiive decision making m problem ms having inhherent multiplee conflicting objectives o wheere targets hav ve been assignned to all the attributes in thhe planning horizon h and wh here decision making m unit is m mainly interested in minimiziing the non-ach hievement of thhe goals.The ethos of goall programmingg lies in the Simon’s S conceept [2] of satiisfying of objectives. GP has h a robust tool for multi objeective decisionn analysis. It appears a to be an appropriatee, powerful, annd appeared as flexible tecchnique in operations researcch for decision making probleems with multiiple conflictingg objectives. Thhe literature on o goal program mming has trem mendously grown. multi criteria deecision makingg (MCDM) app proach. The idea Goal progrramming is perrhaps the most widely used m of GP can be visualized from the conccept of efficienncy introduced by Koopmanss [3] in the con ntext of resourrce allocation planning. Thee roots of goal programmingg lie in the stuudy of Charness, Cooper and Ferguson [4] in w they deaal with executiive compensattion methods. In 1961, Charrnes and Coop per [5] offeredd a 1955 in which, more expliicit definition and a coined the term ‘goal proogramming’. Thereafter,, a large number of studies have been madee by pioneer reesearchers and the significantt methodologiccal developmeent of goal pro ogramming haave been achieeved by Ijiri [66], Lee [7], Iggnizio [8], Scchniederjans [99], Romero [110], Schniederj rjans [11] and other researchhes. The vast literature of ggoal programm ming reflects its i theoretical elegance and significance. In 1980, Narasimhan N [12] employedd the concept of fuzzy set theory introdduced by Zadeeh [13] in gooal programmiing by incorpoorating fuzzy goals g and consstraints withinn the traditionaal goal program mming model in order to addd new dimenssion in modeliing flexibility and accuracy to t the goal proorgramming model m for dealinng with uncerrtainty. Thereaafter, fuzzy goaal progammingg has been furtther developedd by Hannan [114], Ignizio [155], Tiwari et al. a [16, 17], Mohamed [18], Pramanik P and Roy [19, 20], Pramanik andd Dey [21,], Praamanik [22] annd other reseaarchers. Atanassov [23, 24] incorrporated the degree d of non-m membership (rrejection) as ann independent component annd defined inttuitionistic fuzzy set to deal uncertainty u in more flexible way. In 1995 Angelov [25] presented a neew

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Global JJournal of f EngineeringScience and RessearchManagement a t concept to o optimizationn problem inn intuitionisticc fuzzy envirronment. In 11997, Angelov v [26] restateed intuitionisttic fuzzy linear programmiing problem [25] consideriing maximizinng membershiip function annd minimizingg the non-meembership funnctions simulltaneously by extending fuuzzy linear multim objectivve programmm ming proposed d by Zimmerm mann [27]. In 2001, Angelov [28] also presented a generaal formulation of the optimizzation problem m of an air condditioning system m in the framew work of intuitiionistic fuzzy set s theory. uitionistic fuzzzy goal program mming. In 20005, Goal progrramming in inttuitionistic fuzzy environmennt is called intu Pramanik and Roy [29]] proposed inttuitionistic fuzzzy goal proggramming (IFG GP) by extendding fuzzy gooal programmiing. Pramanik and Roy [30, 31, 32] also ppresented intuittionistic fuzzy goal programm ming for qualiity control prooblem, transpo ortation problem ms and bi-leveel programmin ng problems reespectively butt these problem ms are numerrical problems.. Major success has not been achieved in n intuitionisticc multi-objectiive optimizatioon problems. Smarandacche [33, 34, 355, 36] introducced the conceppt of the degreee of indeterminnacy/neutralityy as independeent componentt in 1998 andd defined the neutrosophic set in order to t deal with uuncertainty andd indeterminaccy involved inn real world problems. The significance of Smarandachee’s work [33] is that it is cap pable of dealinng with indeteerminacy which is beyond thee scope of fuzzzy set and intuiitionistic fuzzyy set. The needd of neutrosophhic set was feelt and actuallyy discovered by b Smarandachhe in 1995 annd he wrote thhe manuscript in 1995 but he h published it in 1998. When W the new paradigm was grounded by y Smarandachee [33], the usu ual process off a paradigm shift s started. The T concept off neutrosophicc set, derived from f neutrosopphy, which un nderlies the neew paradigm, was initially ig gnored, ridiculled, or attackedd by many [37 7, 38], while it was supported d only by a veery few, mostly young, unknnown, and uninnfluential reseaarchers. Inspite of the initial lack of interest, skepticism [337, 38], or op pen hostility, the t new paraddigm persevereed with virtuaally no supporrt in the 19900s. Smarandachhe becomes th he torchbearer of neutrosoph hy, neutrosophiic set and neutrrosophic logic. He has tried his level best to propagate the new parad digm by writinng books, e-boooks, providingg the free dow wnloads of his writings in free journals an nd websites. The T new parad digm matured significantly and a gained som me supports inn the 2010s annd started to demonstrate d itss superior praggmatic utility inn the 2010s. The T paradigm sshift initiated by b the concept of neutrosophhy [33] and neutrosophic n seet and the ideea of mathemaatics based onn neutyrosophiic set, which is currently ongoing, o possesses similar charactewristics to other paaradigm shift recognized in n the history of science. The T new paraddigm shift covvers a broad rrange of subjeects, from phillosophy to maathematics. Thhe paradigm shift s is still onggoing and it seeems that it willl probably takee much longer time t than usuaal to complete it. i This can be b concluded because b of thee fact that thee scope of the paradigm shift is very wid de and open annd competitivve. W et al. [39 9] defined singge valued neutrrosophic set (S SVNS) which is an instance of neutrosophhic In 2010, Wang set, whose truth memberrship degree, inndeterminacy aand falsity deggrees lie in thee unit interval [0, [ 1]. It can be b stated that an important point p of evolutiion of the moddern concept off uncertainty w was the publicattion of a seminnal marandache [33]. Although mathematics m baased on SVNSs has far greatter expressive power p than crisp work of Sm set, fuzzy sets, intuitionnistic fuzzy seets, its usefulnness depends critically on one’s capabiliity to formulaate appropriatee membership functions, inddeterminate funnctions and fallsity functions for various giiven concepts in various conntexts and theiir multiple opeerational rules.. Union and inntersections of two SVNSs caan be differenttly defined andd different resu ults can be obtaained for the saame optimizatiion problem. o the theory of o SVNSs has been growing steadily since its inception iin 2010. The body b of conceppts Research on and resultss pertaining to the theory of SVNS S is now iimpressive. Reesearch on a broad variety of applications has h also been very v attractive and has produ uced results thaat are perhaps even e more imppressive [40, 41 1, 42, 43, 44, 45, 4 46, 47]. In 2015, Roy R and Das [48] presented multi-objective m production plaanning problem m based on neuutrosophic lineear programmiing approach. Das and Rooy [49] preseented multi-obbjective non-liinear program mming based on o neutrosophhic optimizatio on technique and a its applicaation in riser design problem m. Hezam et al. [50] studieed Taylor seriies approximattion to solve neeutrosophic muulti-objective programming p pproblem. In 20016, Abdel-Basset et al. [51] proposed p neutrrosophic goal programming p u using deviation n variables. In tthe studies [48, 49, 50, 51], thhe researcherss maximize inddeterminacy. But B in a real maanagement systtem, decision m making unit do oes not show anny interest to maximize inddeterminacy. Because B maxim mization of in ndeterminacy does d offer anyy benefit to thhe

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Global JJournal of f EngineeringScience and RessearchManagement a t managemeent system and d the organizattion. So it is not pragmatic to maximize indeterminacyy function in thhe process off optimizing off the objective functions of thhe decision maaking problemss. So, the techn niques presenteed in the papeers [48, 49, 50,, 51] are neutroosophic in natuure. Their apprroaches went inn wrong directiions. The claim ms of getting better optimall solutions in the t studies [488, 49] are thereefore not validd. However, they initiated neew i i indeterminacy. The errors coommitted by thhem occur due to the choice of idea in opttimization by incorporating defnitions of intersection n of two neuttrosophicsets. Therefore new w methods forr neutrosophicc multi-objectivve a urgently needed. programmiing problems are Fuzzy goall programmingg and intuitionnistic fuzzy goaal programmin ng have been developed d in orrder to deal wiith uncertaintyy. However, th hese two approoaches are not capable of deaaling with indeeterminacy. It seems, therefoore that in manny environmennts it is more reealistic to endeavor achievingg several objectives simultaneeously involvinng indeterminnacy and incom mpleteness. This T observatiion reflects thhat real world problems havve to be solveed optimally according a to criteria c involvinng indeterminaacy. Consequeently, we mustt acknowledge the presence of several objjectives which h are at least contradictory, conflicting, in ndeterminate aand often non--commensurabble leading to the t developmeent of neutrosopphic optimizattion technique.

w framework off neutrosophic llinear goal proogramming model. This paperr develops new Rest of thhe paper has been b organizedd in the follow wing way. Seection 2 presennts some basiic definitions of neutrosophhic sets, Sectiion 3 is devooted to presennt the proposeed frameworkk of neutrosopphic linear gooal programmiing and intuitionistic fuzzy goal g programm ming models. Section S 4 preseents the concluusion and futuure direction of o research worrk.

PRELIM MINARIES We recall some s basic deffinitions related d to neutrosophhic sets which are important tto develop the paper. 2.1 Definittion of neutrosophic set [33] Let V be a space of pointts (objects) with h a generic eleement v ∈ V. A neutrosophic set S in V is ch haracterized byy a mbership functiion TS ( v) , an indeterminacyy membershipp function I S (v) , and a falssity membershhip truth mem function FS (v) and is den noted by S = { v , TS ( v ), I S ( v ), FS ( v ) ⏐v ∈ V.} Here TS ( v ) , I S (v) and FS (v) have beenn defined as follows: −

TS : V →] 0, 1+ [ −

I S : V→] 0, 1+ [ −

FS : V →] 0, 1+ [ −

Here, TS ( v ) , I S ( v) and FS ( v) are the real r standard annd non-standarrd subset of] 0, 1+ [ . In genneral, there is no n restriction on TS ( v ) , I S (v) and FS ( v) . Therefore, −

0 ≤ Inf TS ( v ) + inf I S (v) +infFS(v) ≤ Sup TS ( v ) + Suup I S (v) +Sup FS(v) ≤ 3+ 2.2. Definiition: Single valued v neutrosophic set [39] Let V be a space of pointts with generic element v∈V. A single valueed neutrosophiic set S in V is characterized by b a truth-meembership funcction TS(v), an n indeterminaccy-membership p function IS (v) and a falssity-membershhip function FS(v), for each h point v in V, TS(v), IS (v)), FS(v)∈[0, 1]], when V is continuous c theen single-valueed neutrosophhic set S can bee written as S = ∫ < TS (v), I G (v), FG (v) > / v, v ∈ V. V

When V is discrete, single-valued neutrrosophic set S can c be written as follows: n

S = ∑ < TS ( v i ), I S ( v i ), FS ( v i ) > ⏐vi, vi∈V i =1

Definition n 2.3 [39]: The complement of o a single valued neutrosophiic set S is denooted by S c and is defined by T c (v) = FS ( v) ; I c ( v ) = 1 − I S ( x ) ; F c (v) = TS (v) S S S

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Global JJournal of f EngineeringScience and RessearchManagement a t Definition n 2.4 [39]: Twoo single valuedd neutrosophic sets P and Q are a equal, writtten as P = Q, iff and only if P ⊆ Q and P ⊇ Q. Definition n 2.5 [52]:The union u of two siingle valued neeutrosophic setts P and Q is a single valued neutrosophic set s R, written as R= P ∪ Q, whose w truth membership, inddeterminacy-m membership andd falsity membbership functionns are related to thosse of P and a Q by TR ( v ) = max ( TP ( v ), TQ ( v )) ; I R ( x ) = minn ( I p ( v), I Q ( v )) ; FR ( x ) = miin ( FP ( v), FQ ( v)) for all v in V. Definition n 2.6 [52]: Thee intersection of o two single vvalued neutrossophic sets P aand Q is a neu utrosophic set R written as R = P ∩Q, whose w truth meembership, inddeterminacy-meembership andd falsity memb bership functionns a Q by TR ( v) = min ( TP ( v), TQ ( x )) ; I R ( v ) = maxx ( I P ( v), I Q ( x ))) ; are related to thosse of P and FR ( x ) = maax ( FP ( v ), FQ ( v )) for all v in V.

Definition n 2.7 [52]: Assuume that { Pj : j∈J} be an arbitrary family of o single valuedd neutrosophic sets in V, thenn

∪Pj

mayy be defined as follows: ∪Pj =

v, ∨ TPj (v), ∧ IPj (v), ∧ FP j (v) j∈ J

j∈ J

j∈J

(ii) ∩ P j may m be defined as follows: ∩ P j = v, ∧ TPi (v), ∨ IPj (v), ∨ FPi (v) j∈ j

j∈ J

FORMU ULATION OF O NEUTR ROSOPHIC LINEAR GOAL G PRO OGRAMMIN NG To formullate neutrosophhic goal programming, we start from multi-objective m programing problem p in crisp environmeent. Consider an a optimizationn problem of th he form in crispp environment:: Max Φ i ( v) , i = 1, 2, …, r1 (1) Subject to ψ i ( v ) ≤ 0, i = r1+1, …, r v≥0 where Φ i ( v) represents the i-th objecttive function, v is the vector of decision vaariables (v1,v2, …, vk ), ψ i ( v ) denotes i-thh constraint, r denotes the nuumber of objecttive functions and a s denotes tthe number of constraints.

Analogouss fuzzy optimiization problem In general,, fuzzy optimizzation problem m comprises off a set of objecctives and constraints. The objectives o and or constraintss or parameterss and relationss are expressedd by fuzzy setss which explaiin the degree of o satisfaction of the respecttive condition and a expressed by their membbership function ns [53]. Consider thhe analogous fuzzy f optimizattion problem: ~

Max Φ i ( v) , i = 1, 2, …,, r1

(22)

Subject to ~

0 i = r1+1, …, r ψ i ( v ) ≤ 0, v≥0 ~

Μax denottes fuzzy maxiimization and ≤ denotes the fuzzy inequaliity. To maximiize the degree of membershipp of the objectiives and constrraints to the resspective fuzzy sets: Max μi( v ), ) v ∈ℜk, i = 1,, 2, …, r1, r1+1,, …, r Subject to … r1, r1+1, …, r 0 ≤μi( v ) ≤ 1, i = 1, 2, …,

(33)

v≥0

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Global JJournal of f EngineeringScience and RessearchManagement a t Where μi( v ) denotes th he degree of membership m off i-th objectivee function Φ i ( v) (i = 1, 2, …r1) and μi( v ) denotes thee degree of i-thh membership function of coonstraint ψ i ( v ) (i = r1+1, …, r ). Minimum operator of Beellman and Zad deh [54] can bee applied to thee optimization pproblem (3). r

μ D( v ) = ∧ μ i ( v ) , v ≥ 0 , i = 1, 2, …, r1, r1+1, …, r

(44)

Therefore, μ D( v )≤ μ i ( v ) , i = 1, 2, …, r1, r1+1, …, r s as follow ws: According to Zimmermannn [55], the problem can be solved

(55)

i =1

μ D ( v ) = Max (min ( μ 1 ( v ) , μ 2 ( v ) , …, … μ r1 ( v) , μ r1+1 ( v) , …, μ r ( v ) )

(66)

Subject to 0 ≤μi( v ) ≤ 1, i = 1, 2, …, r1, r1+1, …, r v≥0 . m: The probleem (6) is equivalent to the folllowing problem Max α α≤μi( v ), i = 1, 2, …, r1, r1+1, …, r

(77)

v≥0 .

gous intuitioniistic fuzzy optiimization (IFO O) problem An analog An analogoous intuitionisttic fuzzy optim mization probleem can be repreesented as folloows: To maxim mize the degreee of acceptannce of intuitioonistic fuzzy objective o funcctions and connstraints, and to minimize the t degree of reejection of intuuitionistic fuzzyy objective fun nctions and connstraints we caan write: Max μi( v ), ) v ∈ℜk , i = 1, 2, …, r1, r1+1 1, …, r (88) Min νi( v ), v ∈ℜk, i = 1,, 2, …, r1, r1+1,, …, r (99) Subject to μi( v ) + νi( v ) ≤ 1 i = 1, 2, 2 …, r1, r1+1, …, r, νi( v ) ∈ [00, 1], i = 1, 2, …, … r1, r1+1, …,, r, μi( v ) ∈ [0 0, 1], i = 1, 2, …, … r1, r1+1, …, r, v≥0 d of memb bership of i-th objective funcction Φ i ( v) (i = 1, 2, …r1) an nd μi( v ) denottes Here μi( v ) denotes the degree the degree of i-th membeership functionn of constraint ψ i ( v ) (i = r1+1, …, r). Here νi( v ) denotes the degree of nonn-membership of i-th objective function Φ i ( v) (i = 1, 2,, …r1)and νi( v ) denotes thee degree of i-thh non-memberrship function oof constraint ψ i ( v ) (i = r1+1,, …, r). Conjunctioon of intuitionistic fuzzy sets can be definedd as follows: G ∩C = {〈 v , μG( v ) ∧ μC( v ),νG( v ) ∨ νC( v )〉| v ∈ℜk}, (10) where G reepresents an in ntuitionistic fuzzzy objectives and a C represen nts constraints. This conjuncttion operator caan be easily generalized g and d applied to thee IFO problem.. Here, r

i =1

D = {〈 v , μD( v )),νD( v )〉| v ∈ℜk}, μD( v ) = ∧ μi( v ),νD( v ) = ∨ νi( v )

(11)

where D reepresents an inttuitionistic fuzzzy set based reepresentation of o the decision. Min-operattor can be usedd for conjunctioon and max-opperator for disju unction. r

μD( v ) = ∧ μi( v ), v ∈ℜk, i = 1, 2, …, r1,1 r1+1, …, r, i =1 r

νD( v ) = ∨ νi( v ), v ∈ℜk, i = 1, 2, …, r1, r1+1, …, r, i =1

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(122) (13)

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Global JJournal of f EngineeringScience and RessearchManagement a t Therefore, μD( v ) ≤μi( v ),νD ( v ) ≥νi( v ), ) i = 1, 2, …, r1, r1+1, …, r,

(14))

mization problem can be transformed iinto intuitionistic fuzzy gooal The abovee intuitionisticc fuzzy optim programmiing problem ass follows: To maximize m the ddegree the acceptance of intuuitionistic fuzzzy objectives annd constraintss, and to minim mize the degreee of rejection off intuitionistic objectives andd constraints, we w can write k (15) Max μi( v ), ) v ∈ℜ , i = 1,, 2, …, r1, r1+1,, …, r, (166) Min νi( v ),, v ∈ℜk, i = 1, 2, …, r1, r1+11, …, r , Subject to νi( v ) ≥ 0, i = 1, 2, …, r1, r1+1, …, r, μi( v ) + νi( v ) ≤ 1 i = 1, 2, 2 …, r1, r1+1, …, r, v≥ 0 .

b For the deffined membersship function μi( v ), the flexibble membershiip goals havingg the aspired leevel unity can be presented as a follows: μi( v ) + d −i1i −d +i1 = 1, i = 1, 2, …, r1, r1+1, + …, r (177) For the casse of rejection (non-membersship), we can w write − + νi( v ) + d i2i −d i2 = 0, i = 1, 2, …, r1, r1+1, + …, r

(188)

Since decision making unit u wants to minimize m the deegree of rejecttion and maxim mize the degreee of acceptancce, IFGP can be b formulated as: a IFGP moddel-1 Min λ Subject to + i1

=1, i = 1, 2, …, r1, r1+1, …, r,

+ −d i2

= 0, i = 1, 2, …, r1, r1+1, + …, r,

μi( v ) + d ‐ i1i - d νi( v ) +

− d i2 i

(199)

μi( v ) + νi( v ) ≤ 1 i = 1, 2, 2 …, r1, r1+1, …, r,

λ ≥ d −i1 , i = 1, 2, …, r1, r1+1, …, r, λ ≥ d +i2 , i = 1, 2, …, r1, r1+1, …, r,

d−i1 × d +i1 = 0, i = 1, 2, …, r1, r1+1, …, r,, d −i2 × d +i2 = 0, i = 1, 2, …, r1, r1+1, …, r, r d−i1 ≥ 0, d +i1i ≥ 0, d−i2 ≥ 0, d+i2 ≥ 0, i = 1, 2, 2 …, r1, r1+1, … …, r, v≥0 .

a) Model (IIa The minim mization of the sum of the weiighted deviatioon form: r

i =1

Min η = ∑ w i1- d i1- + ∑ w i2i+ d i2+

(20)

Subject to μi( v ) + d−i1i - d +i1 =1, i = 1, 1 2, …, r1, r1+1, …, r, − + νi( v ) + d i2i − d i2 = 0, i = 1, 2, …, r1, r1+1, + …, r,

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Global JJournal of f EngineeringScience and RessearchManagement a t μi( v ) + νi( v ) ≤ 1 i = 1, 2, 2 …, r1, r1+1, …, r,

d−i1 × d +i1 = 0, i = 1, 2, …, r1, r1+1, …, r,, d −i2 × d +i2 = 0, i = 1, 2, …, r1, r1+1, …, r, r … r, w i1- ≥ 0, w i2+ ≥ 0, i = 1, 2,, …, r1, r1+1, …,

d−i1 ≥ 0, d +i1i ≥ 0, d−i2 ≥ 0, d+i2 ≥ 0, i = 1, 2, 2 …, r1, r1+1, … …, r, v≥0 .

b) Model (IIb The minim mization of the sum of the devviation form: r

i =1

Min ζ = ( ∑ d i1- + ∑ d i2+ )

(21)

Subject to the same set off constraints (2 20). +

d varriables. The num Here, d −i1 , and d +i2 , are deviational merical weightts w i1 , w i2 asssociated with d −i1 , d +i2 represeent o achieving th he aspired leveel of the respeective intuitionnistic fuzzy goal subject to thhe the relativee importance of given set of o constraints. To T assess the relative r importaance of the intuuitionistic fuzzzy goals, the weighting w schem me -

suggested by b Pramanik and a Roy [29] caan be used to aassign the valuees of w i1 , w i2 . Formulatiion of the neuttrosophic goall programmin ng Neutrosoph hic optimizatio on problem can n be representeed as follows: To maximiize the degree of acceptance (truth) of neutrrosophic objecctives and consstraints, to miniimize the degree of indeterm minacy and to minimize m the degree d of rejecttion (falsity) off neutrosophic objectives andd constraints: k (222) Max μi( v ), ) v ∈ℜ , i = 1, 1 2, …, r1, r1+11, …, r, Min ωi( v ), ) v ∈ℜk, i = 1, 2, …, r1, r1+1, …, r, Min νi( v ), v ∈ℜk, i = 1, 2, …, r1, r1+1, …, r, Subject to μi( v ) + ωi( v ) + νi( v ) ≤ 3, i = 1, 2, …, … r1, r1+1, …, rr, μi( v )∈[0, 1], i = 1, 2, …, … r1, r1+1, …, r, r ωi( v )∈[0,, 1], i = 1, 2, …, … r1, r1+1, …, r, r νi( v )∈[0, 1], i = 1, 2, …, … r1, r1+1, …, r, r v≥0 . where μi( v ) denotes thhe degree of membership m o v to the i-thh SVNS and νi( v ) denotees the degree of of rejection of functions v from the i-th SVNS. S Conjunctioon of SVNSs iss defined by G ∩C = {〈 v , μG( v ) ∧ μC( v ),ωG( v ) ∨ ωC( v ), νG( v ) ∨ νC( v )〉| v ∈ℜk}, (23) Here G reppresents a neuttrosophic objecctive function and C represennts neutrosophhic constraint. This T conjunctioon operator caan be easily genneralized and applied a to the neutrosophic n optimization o prooblem: r

i =1

D = {〈 v , μD( v )),νD( v )〉〉| v ∈ℜk}, μD( v ) = ∧ μi( v ),ω ωD( v )= ∨ ωi ( v ) , r

νD( v ) = ∨ νi( v )

(244)

i =1

where D reepresents a singgle valued neutrosophic set bbased representtation of the deecision. Min-operattor is used for conjunction an nd max-operatoor for disjunctiion: μD( v )

= ∧ μi( v ), i =1

v ∈ℜk,

ωD( v )

= ∨ ωi( v ), v ∈ℜk, i =1

νD ( v )

∨ νi( v ),

i =1

v ∈ℜ ℜ k.

(25)

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Global JJournal of f EngineeringScience and RessearchManagement a t Therefore, μD( v ) ≤μ μi( v ),ωD( v ) ≥ωi( v ), νD( v ) ≥νi( v ),i ) = 1, 22, …, r1, r1+1, …, r. (26) where μi( v ) denotes the degree off membership of v to the i-th SVNS, ωi( v ) denotess the degree of indeterminnacy, and νi( v ) denotes the degree d of rejection of functions v from thee i-th SVNS. del (I). NGP Mode Minimize λ Subject to μi( v ) + d −i1 - d +i1 =1, i = 1, 2, …, r1, r1+1, + …, r,

(27)

+ ωi( v ) + d -i2 - d i2 = 0, i = 1, 2, …, r1, r1+1, …, r,

νi( v ) + d -i3 - d +i3 = 0, i = 1, 2, …, r1, r1+1, …, r,

λ ≥ d −i1 , i = 1, 2, …, r1, r1+1, …, r, λ ≥ d +i2 , i = 1, 2, …, r1, r1+1, …, r, λ ≥ d +i3 , i = 1, 2, …, r1, r1+1, …, r, μi( v ) + ωi( v ) + νi( v ) ≤ 3, i = 1, 2, …, … r1, r1+1, …, r, r

d −i1 ≥ 0, d -i2 ≥ 0, d -i3 ≥ 0, 0 i = 1, 2, …, r1, r1+1, …, r, d −i1 × d +i1 = 0, i = 1, 2, …, r1, r1+1, …, r, d -i2 × d +i2 = 0, i = 1, 2, …, … r1, r1+1, …, r, r d -i3 × d +i3 = 0, i = 1, 2, …, … r1, r1+1, …, r, μi( v )∈[0, 1],

i = 1, 2, 2 …, r1, r1+1, …, … r,

ωi( v )∈[0, 1], i = 1, 2, …, … r1, r1+1, …, r, νi( v )∈[0, 1], i = 1, 2, …, … r1, r1+1, …, r, r

v≥0 . NGP Mode del (IIa) r

i =1

Min η = ∑ w i1- d i1- + ∑ w i2+ d i2+ + ∑ w i3+ d i3+

(28)

Subject to − μi( v ) + d i1 - d +i1 =1, i = 1, 2, …, r1, r1+1, + …, r, + ωi( v ) + d -i2 - d i2 = 0, i = 1, 2, …, r1, r1+1, …, r,

νi( v ) + d -i3 - d +i3 = 0, i = 1, 2, …, r1, r1+1, …, r, μi( v ) + ωi( v ) + νi( v ) ≤ 3, i = 1, 2, …, … r1, r1+1, …, r, r

d −i1 × d +i1 = 0, i = 1, 2, …, r1, r1+1, …, r, d -i2 × d +i2 = 0, i = 1, 2, …, … r1, r1+1, …, r, r d -i3 × d +i3 = 0, i = 1, 2, …, … r1, r1+1, …, r,

d −i1 ≥ 0, d +i1 ≥ 0, d -i2 ≥ 0, d +i2 ≥ 0, d -i3 ≥ 0, d +i3 ≥ 0, i = 1, 2, …, r1, r1+1, + …, r, w i1- ≥ 0, w i2+ ≥ 0, w i3+ ≥ 0, i = 1, 2, …, r1,1 r1+1, …, r,

μi( v )∈[0, 1], i = 1, 2, …, r1, r1+1, …, r, r ωi( v )∈[0, 1], i = 1, 2, …, … r1, r1+1, …, r,

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Global JJournal of f EngineeringScience and RessearchManagement a t νi( v )∈[0, 1], i = 1, 2, …, … r1, r1+1, …, r, r

v≥0 . NGP Modeel (IIb). r

(29)) r

+ i2

Min η = ∑ d + ∑ d + ∑ d i =1

i =1

+ i3

Subject to the same set off constraints (2 28). Here d

− i1

, d -i2 , d -i3 , d +i1 , d i2 , d +i3 are deviational d varriables. The nu umerical weighhts wi1 , wi2 , wi33 associated wiith -

d −i1 , d +i2 , d +i3 represent the relative im mportance of aachieving the aspired level oof the respectiive neutrosophhic goal subject to the givenn set of constrraints. To asseess the relativee importance oof the neutrosophic goals, thhe weighting scheme suggessted by Pramannik and Roy [229] can be usedd to assign the vvalues of wi1- , wi2+ , wi3+ .

CONCL LUSION This paperr presents fram mework of neutrosophic linnear goal progrramming probblem. Three neew intuitionisttic fuzzy goall programming g models havee been presentted. The proposed intuitioniistic fuzzy goal programminng models havve been also ex xtended to neuutrosophic linear goal program mming modelss. The essence of the proposeed neutrosophhic linear goaal programmin ng is that it is capable of o dealing witth indeterminacy and falsiity simultaneoously. Abdel-B Baset et al. [51] presented gooal programminng models in 22016. Howeveer, in their studdy they maxim mize indetermiinacy which iss not realistic in i decision maaking context. In this paper the t definition of intersection n of two singlle valued neuttrosophic sets due to Salamaa and Alblowii [52] has beenn employed annd direction of o research in neutrosophic optimization problem has been proposed. The authorr hopes that thhe proposed framework f of neutrosophic n liinear goal proggramming will open up new avenue of reseearch in the fieeld of optimizzation problem ms in neutrosophic environm ment. Many arreas need to bbe explored annd developed in neutrosophhic goal prog gramming especially prioritty structure of o neutrosophhic goals and priority baseed neutrosophhic linear goal programming. p

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