Neutrosophic Linear Programming Problems

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1 Neutrosophic Operatioal Research I Neutrosophic Liear Programmig Problems Abdel-Nasser Hussia Mai Mohamed Mohamed Abdel-Baset 3 Floreti Smaradache 4 Departmet of Iformatio System, Faculty of Computers ad Iformatics, Zagazig Uiversity, Egypt. asserhr@gmail.com,3 Departmet of Operatios Research, Faculty of Computers ad Iformatics, Zagazig Uiversity, Sharqiyah, Egypt. E_mai0@yahoo.com aalyst_mohamed@yahoo.com 4 Math & Sciece Departmet, Uiversity of New Mexico, Gallup, NM 8730, USA. smarad@um.edu Abstract Smaradache preseted eutrosophic theory as a tool for hadlig udetermied iformatio. Wag et al. itroduced a sigle valued eutrosophic set that is a special eutrosophic sets ad ca be used expedietly to deal with real-world problems, especially i decisio support. I this paper, we propose liear programmig problems based o eutrosophic eviromet. Neutrosophic sets are characterized by three idepedet parameters, amely truthmembership degree (), idetermiacy-membership degree (I) ad falsity-membership degree (F), which are more capable to hadle imprecise parameters. We also trasform the eutrosophic liear programmig problem ito a crisp programmig model by usig eutrosophic set parameters. o measure the efficiecy of our proposed model we solved several umerical examples. Keywords Liear Programmig Problem; Neutrosophic; Neutrosophic Sets. Itroductio Liear programmig is a method for achievig the best outcome (such as maximum profit or miimum cost) i a mathematical model represeted by liear relatioships. Decisio makig is a process of solvig the problem ad achievig goals uder asset of costraits, ad it is very difficult i some cases due to 5

2 Editors: Prof. Floreti Smaradache Dr. Mohamed Abdel-Basset Dr. Yogqua Zhou icomplete ad imprecise iformatio. Ad i Liear programmig problems the decisio maker may ot be able to specify the objective fuctio ad/or costraits fuctios precisely. I 995, Smaradache [5-7] itroduce eutrosophy which is the study of eutralities as a extesio of dialectics. Neutrosophic is the derivative of eutrosophy ad it icludes eutrosophic set, eutrosophic probability, eutrosophic statistics ad eutrosophic logic. Neutrosophic theory meas eutrosophy applied i may fields of scieces, i order to solve problems related to idetermiacy. Although ituitioistic fuzzy sets ca oly hadle icomplete iformatio ot idetermiate, the eutrosophic set ca hadle both icomplete ad idetermiate iformatio. [,5-7] Neutrosophic sets characterized by three idepedet degrees amely truthmembership degree (), idetermiacy-membership degree(i), ad falsitymembership degree (F), where,i,f are stadard or o-stadard subsets of ]0 -, + [. he decisio makers i eutrosophic set wat to icrease the degree of truth-membership ad decrease the degree of idetermiacy ad falsity membership. he structure of the paper is as follows: the ext sectio is a prelimiary discussio; the third sectio describes the formulatio of liear programig problem usig the proposed model; the fourth sectio presets some illustrative examples to put o view how the approach ca be applied; the last sectio summarizes the coclusios ad gives a outlook for future research. Some Prelimiaries. Neutrosophic Set [] Let X be a space of poits (objects) ad x X. A eutrosophic set A i X is defied by a truth-membership fuctio (x), a idetermiacy-membership fuctio (x) ad a falsity-membership fuctio (x). (x), I A (x) ad F A (x) are real stadard or real ostadard subsets of ]0 -, + [. hat is A (x):x ]0 -, + [, I A (x):x ]0,+[ ad F A (x):x ]0 -, + [. here is o restrictio o the sum of (x), I A (x) ad F A (x), so 0 - A (x) supi A (x) F A (x) Sigle Valued Neutrosophic Sets (SVNS) [7,8] Let X be a uiverse of discourse. A sigle valued eutrosophic set A over X is a object havig the form A = { x, A (x), I A (x), F A (x) :x X}, where A (x):x [0,], I A (x):x [0,] ad F A (x):x [0,] with 0 A (x)+ I A (x)+ F A (x) 3 for all x X. he itervals (x), I A (x)ad F A (x) deote the truth-membership degree, the idetermiacymembership degree ad the falsity membership degree of x to A, respectively. 6

3 Neutrosophic Operatioal Research For coveiece, a SVN umber is deoted by A= (a, b, c), where a, b, c [0, ] ad a+b+c 3..3 Complemet [3] he complemet of a sigle valued eutrosophic set A is deoted by C (A) ad is defied by c (A)(x) = F(A)(x), I c (A)(x) = I(A)(x), F c (A)(x) = (A)(x), for all x i X..4 Uio [3] he uio of two sigle valued eutrosophic sets A ad B is a sigle valued eutrosophic set C, writte as C = A B, whose truth-membership, idetermiacy membership ad falsity-membership fuctios are give by (C)(x) = max ( (A)(x), (B)(x) ), I(C)(x) = max (I(A)(x), I(B)(x)), F(C)(x) = mi( F(A)(x), F(B)(x) ) for all x i X..5 Itersectio [3] he itersectio of two sigle valued eutrosophic sets A ad B is a sigle valued eutrosophic set C, writte as C = A B, whose truth-membership, idetermiacy membership ad falsity-membership fuctios are give by (C)(x) = mi ( (A)(x),(B)(x) ), I(C)(x) = mi ( (A)(x),I(B)(x) ), F(C)(x) = max( F(A)(x),F(B)(x) ) for all x i X 3 Neutrosophic Liear Programmig Problem Liear programmig problem with eutrosophic coefficiets (NLPP) is defied as the followig: Maximize Z= Subject to j= c j x j j= a ij x j b i i m () x j 0, j where a ij is a eutrosophic umber. 7

4 Editors: Prof. Floreti Smaradache Dr. Mohamed Abdel-Basset Dr. Yogqua Zhou he sigle valued eutrosophic umber (a ij ) is doated by A=(a,b,c) where a,b,c [0,] Ad a,b,c 3 defied as: he truth- membership fuctio of eutrosophic umber a ij is defied as: x a a a a x a a ij (x) = { a x a x a 3 a 3 a 0 otherwise he idetermiacy- membership fuctio of eutrosophic umber a ij is () defied as: I a ij (x) = { x b b b b x b b x b 3 b 0 otherwise b x b 3 Ad its falsity- membership fuctio of eutrosophic umber a ij (3) is F a ij (x) = x C C C C x C C x C 3 C { otherwise C x C 3 he we fid the upper ad lower bouds of the objective fuctio for truth-membership, idetermiacy ad falsity membership as follows: z U = max{z(x i )} ad z l =mi{z(x i )} where i k z F L= z L Ad z F u= z u R(z u z L ) I z U= I z U I ad z l= I S(z u z L ) z l= where R, S are predetermied real umber i (0, ). he truth membership, idetermiacy membership, falsity membership of objective fuctio are as follows: O (Z) = { z z L z u z L if z z u if z L z z u 0 if z < z L if z z u z z L I (Z) I O = z I u zi if z L z z u L { 0 if z < z L (4) (5) (6) 8

5 Neutrosophic Operatioal Research { F O (Z) = { if z z u z u F Z z u F z L F if z L z z u 0 if z < z L he eutrosophic set of the i th (x) ci = if b i j= (a ij + d ij )x j b i j= a ij xj j= d ij xj costrait c i is defied as: if j= a ij xj b i < j= (a ij + d ij )x j 0 if b i < j= a ij xj (x) I ci = F (x) ci = { 0 if b i < a ij xj b i j= d ij xj j= a ij xj j= { 0 if b i j= (a ij + d ij )x j If j= a ij xj b i < j= (a ij + d ij )x j if b i < j= a ij xj (x) ci if j= a ij xj b i < j= (a ij + d ij )x j 0 if b i j= (a ij + d ij )x j (7) (8) (9) (0) 4 Neutrosophic Optimizatio Model I our eutrosophic model we wat to maximize the degree of acceptace ad miimize the degree of rejectio ad idetermiacy of the eutrosophic objective fuctio ad costraits. Neutrosophic optimizatio model ca be defied as: mif (x) mii (x) max (x) Subject to (X) F (x), (X) I (x), 0 (X) + I (x) + F (x) 3, () (X), I (X), F (X) 0, x 0, 9

6 Editors: Prof. Floreti Smaradache Dr. Mohamed Abdel-Basset Dr. Yogqua Zhou where (x), F (x), I (x) deote the degree of acceptace, rejectio, ad idetermiacy of x respectively. he above problem is equivalet to the followig: max α, mi β, mi θ Subject to α (x), β F (x), θ I (x), α β, α θ, 0 α + β + θ 3, () x 0, where α deotes the miimal acceptable degree, β deotes the maximal degree of rejectio ad θ deotes the maximal degree of idetermiacy. he eutrosophic optimizatio model ca be chaged ito the followig optimizatio model: max(α β θ) Subject to α (x), (3) β F (x), θ I (x), α β, α θ, 0 α + β + θ 3, α, β, θ 0, x 0. he previous model ca be writte as: mi (- α) β θ Subject to 0

7 Neutrosophic Operatioal Research α (x) β F (x) θ I (x) α β α θ 0 α + β + θ 3 (4) x 0. 5 he Algorithm for Solvig Neutrosophic Liear Programmig Problem (NLPP) Step. solve the objective fuctio subject to the costraits. Step. create the decisio set which iclude the highest degree of truthmembership ad the least degree of falsity ad idetermiacy memberships. Step 3. declare goals ad tolerace. Step 4. costruct membership fuctios. Step 5. set α, β, θ i the iterval ] - 0, + [ for each eutrosophic umber. Step 6. fid the upper ad lower boud of objective fuctio as we illustrated previously i sectio 3. Step 7. costruct eutrosophic optimizatio model as i equatio (3). 6 Numerical Examples o measure the efficiecy of our proposed model, we solved may umerical examples. 6.. Illustrative Example # Beaver Creek Pottery Compay is a small crafts operatio ru by a Native America tribal coucil. he compay employs skilled artisas to produce clay bowls ad mugs with authetic Native America desigs ad colours. he two primary resources used by the compay are special pottery clay ad skilled labour. Give these limited resources, the compay desires to kow how may bowls ad mugs to produce each day i order to maximize profit. he two products have the followig resource requiremets for productio ad profit per item produced preseted i able :

8 Editors: Prof. Floreti Smaradache Dr. Mohamed Abdel-Basset Dr. Yogqua Zhou able. Resource requiremets of two products product Resource Requiremets Labour(Hr./Uit) Clay (Lb./Uit) Profit($/Uit) Bowl 4 40 Mug 3 50 here are aroud 40 hours of labour ad aroud 0 pouds of clay available each day for productio. We will formulate this problem as a eutrosophic liear programmig model as follows: where max 40 x + 50 x S.t. x + x 40 4 x + 3 x 0 x, x 0 (5) C =40 = {(30, 40, 50), (0.7, 0.4, 0.3)}; C=50 = {(40, 50, 60), (0.6, 0.5, 0.)} ; a = = {(0.5,, 3), (0.6, 0.4, 0.)} ; a = = {(0,, 6), (0.6, 0.4, 0.)} ; a =4 = {(, 4, ), (0.4, 0.3, 0.)} ; a =3 = {(, 3, 0), (0.7, 0.4, 0.3)} ; b =40 = {(0, 40, 60), (0.4, 0.3, 0.5)} ; b =0 = {(00, 0, 40), (0.7, 0.4, 0.3)} ; he equivalet crisp formulatio is: max 5x + 8x S.t x + x 3x + x 45 x, x 0 he optimal solutio is x = 0; x =; with optimal objective value = 6$.

9 Neutrosophic Operatioal Research 6.. Illustrative Example # m a x5 x s. t. 4 x x x, x 3 x 3 x 0 3 x 6 (6) where c = 5= {(4, 5, 6), (0.5, 0.8, 0.3)}; c = 3= {(.5, 3, 3.), (0.6, 0.4, 0)}; a = 4 = {(3.5, 4, 4.), (0.75, 0.5, 0.5)}; a = 3= {(.5, 3, 3.), (0., 0.8, 0.4)}; a = = {(0,, ), (0.5, 0.5, 0)}; a = 3= {(.8, 3, 3.), (0.75, 0.5, 0.5)}; b = = {(,, 3), (0., 0.6, 0.5)}; b = 6 = {(5.5, 6, 7.5), (0.8, 0.6, 0.4)}. he equivalet crisp formulatio is: max.35x x S.t.5375x x x +.5x.375 x, x 0 he optimal solutio is x = ; x = 0; with optimal objective value $. 3

10 Editors: Prof. Floreti Smaradache Dr. Mohamed Abdel-Basset Dr. Yogqua Zhou 6.3. Illustrative Example #3 ma x 5x s. t. 48x 5x 30x where 4x x x, x 6 x 4x (7) c = 5= {(9, 5, 33), (0.8, 0., 0.4)}; c = 48= {(44, 48, 54), (0.75, 0.5, 0)}. he correspodig crisp liear programs give as follows: max.069x +.85x s.t 5x +30x x +6x 4000 x, x 0 he optimal solutio is x = 0; x = 500; with optimal objective value 349 $ 6.4. Illustrative Example #4 ma x 5x s. t. 5x 4x x x, x 30x 6 x 4x 0 48x (8) 4

11 Neutrosophic Operatioal Research where a = a = a = a = a 3 = 5= {(4, 5, 7), (0.75, 0.5, 0.5)}; 30= {(5, 30, 34), (0.5, 0.7, 0.4)}; 4 = {(, 4, 6), (0.4, 0.6, 0)}; 6 = {(4, 6, 8), (0.75, 0.5, 0.5)}; = {(7,, ), (, 0.5, 0)}; a 3 = 4= {(, 4, 9), (0.6, 0.4, 0)}; b = 45000= {(44980, 45000, 45030), (0.3, 0.4, 0.8); b= 4000= {(3980, 4000, 4050), (0.4, 0.5, 0.5)}; b 3 = 8000= {(7990, 8000, 8030), (0.9, 0., 0)}. he associated crisp liear programs model will be: max 5x +48x s.t 5.75x x x x x, x 0 he optimal solutio is x = 0; x =45; with optimal objective value 69648$ 6.5. Illustrative Example#5 max 7x s. t. x 4 x x, x x 3 x 0 5x 6 (9) 5

12 Editors: Prof. Floreti Smaradache Dr. Mohamed Abdel-Basset Dr. Yogqua Zhou where a = = {(0.5,, ), (0., 0.6, 0.3)}; a = = {(.5, 3, 3.), (0.6, 0.4, 0.)}; a = 4 = {(3.5, 4, 4.), (0.5, 0.5, 0.5)}; a = 3= {(.5, 3, 3.), (0.75, 0.5, 0)}; he associated crisp liear programs model will be: max 7x +5x S. t 0.84x +.4x 6.45x +.36x x, x 0 he optimal solutio is x = 4; x =4; with optimal objective value 48$. he result of our NLP model i this example is better tha the results obtaied by ituitioistic fuzzy set [4]. 7 Coclusios ad Future Work Neutrosophic sets ad fuzzy sets are two hot research topics. I this paper, we propose liear programmig model based o eutrosophic eviromet, simultaeously cosiderig the degrees of acceptace, idetermiacy, ad rejectio of objectives, by proposed model for solvig eutrosophic liear programmig problems (NlPP). I the proposed model, we maximize the degrees of acceptace ad miimize idetermiacy ad rejectio of objectives. NlPP was trasformed ito a crisp programmig model usig truth membership, idetermiacy membership, ad falsity membership fuctios. We also give umerical examples to show the efficiecy of the proposed method. As far as future directios are cocered, these will iclude studyig the duality theory of liear programmig problems based o Neutrosophic. Ackowledgemets he authors would like to thak aoymous referees for the costructive suggestios that improved both the quality ad clarity of the paper. 6

13 Neutrosophic Operatioal Research Refereces [] Smaradache, F. A Uifyig Field i Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. Ifiite Study, 005. [] Smaradache, F. A Geometric Iterpretatio of the Neutrosophic Set-A Geeralizatio of the Ituitioistic Fuzzy Set. arxiv preprit math/040450(004). [3] R. Şahi, ad Muhammed Y. A Multi-criteria eutrosophic group decisio makig metod based OPSIS for supplier selectio. arxiv preprit arxiv: (04). [4] Parvathi, R., ad Malathi, C. Ituitioistic fuzzy liear programmig problems. World Applied Scieces Joural. (0): -5. [5] I. M. Hezam, M. Abdel-Baset, F. Smaradache. aylor Series Approximatio to Solve Neutrosophic Multiobjective Programmig Problem. I: Neutrosophic Sets ad Systems. A Iteratioal Joural i Iformatio Sciece ad Egieerig, Vol. 0 (05), pp [6] El-Hefeawy, N., Metwally, M. A., Ahmed, Z. M., & El-Heawy, I. M. A Review o the Applicatios of Neutrosophic Sets. Joural of Computatioal ad heoretical Naosciece, 3(), (06), pp [7] Abdel-Baset, M., Hezam, I. M., & Smaradache, F. Neutrosophic Goal Programmig. I: Neutrosophic Sets & Systems, vol. (06). [8] Abdel-Basset, M., Mohamed, M. & Sagaiah, A.K. J Ambiet Itell Huma Comput (07). DOI: 7