Taylor Series Approximation to Solve Neutrosophic Multi-objective Programming Problem

Transcription

1 Taylor Seres Appromaton to Solve Neutrosophc Mult-objectve Programmng Problem Ecerpt from NETROSOPHC OPERATONA RESEARCH, Volume. Edtors: Prof. Florentn Smarandache, Dr. Mohamed Abdel-Basset, Dr. Yongquan Zhou. Foreword by John R. Edwards. Preface by the edtors. Brussels (Belgum): Pons, 2017, 210 p. SBN brahm M. Hezam 1 Mohamed Abdel-Baset 2 * Florentn Smarandache 3 1 Department of computer, Faculty of Educaton, bb nversty, bb cty, Yemen. E-mal: brahzam.math@gmal.com 2 Department of Operatons Research, Faculty of Computers and nformatcs, Zagazg nversty, Sharqyah, Egypt. E-mal: analyst_mohamed@yahoo.com 3 Math & Scence Department, nversty of New Meco, Gallup, NM 87301, SA. E-mal: smarand@unm.edu Abstract n ths chapter, Taylor seres s used to solve neutrosophc multobjectve programmng problem (NMOPP). n the proposed approach, the truth membershp, ndetermnacy membershp, falsty membershp functons assocated wth each objectve of multobjectve programmng problems are transformed nto a sngle objectve lnear programmng problem by usng a frst order Taylor polynomal seres. Fnally, to llustrate the effcency of the proposed method, a numercal eperment for suppler selecton s gven as an applcaton of Taylor seres method for solvng neutrosophc mult-objectve programmng problem. Keywords Taylor seres; Neutrosophc optmzaton; Mult-objectve programmng problem. 1 ntroducton n 1995, Smarandache [13], startng from phlosophy (when [8] fretted to dstngush between absolute truth and relatve truth or between absolute falsehood and relatve falsehood n logcs, and respectvely between absolute membershp and relatve membershp or absolute non-membershp and relatve non-membershp n set theory) [12], began to use the non-standard analyss. Also, nspred from the sport games (wnnng, defeatng, or te scores), from votes (pro, contra, null/black votes), from postve/negatve/zero numbers, from yes/no/na, 77

2 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou from decson makng and control theory (makng a decson, not makng one, or hestatng), from accepted/rejected/pendng, etc., and guded by the fact that the law of ecluded mddle dd not work any longer n the modern logcs [12], Smarandache combned the non-standard analyss wth a tr-component logc/set/probablty theory and wth phlosophy. How to deal wth all of them at once, s t possble to unfy them? [12]. Netrosophc theory means Neutrosophy appled n many felds n order to solve problems related to ndetermnacy. Neutrosophy s a new branch of phlosophy that studes the orgn, nature, and scope of neutraltes, as well as ther nteractons wth dfferent deatonal spectra. Ths theory consders every entty <A> together wth ts opposte or negaton <anta> and wth ther spectrum of neutraltes <neuta> n between them (.e. enttes supportng nether <A> nor<anta>). The <neuta> and <anta> deas together are referred to as <nona>. Neutrosophy s a generalzaton of Hegel's dalectcs (the last one s based on <A> and <anta> only). Accordng to ths theory every entty <A> tends to be neutralzed and balanced by <anta> and <nona> enttes - as a state of equlbrum. n a classcal way <A>, <neuta>, <anta> are dsjont two by two. But, snce n many cases the borders between notons are vague, mprecse, Sortes, t s possble that <A>, <neuta>, <anta> (and <nona> of course) have common parts two by two, or even all three of them as well. Hence, n one hand, the Neutrosophc Theory s based on the trad <A>, <neuta>, and <anta>. n the other hand, Neutrosophc Theory studes the ndetermnacy, labeled as, wth n = for n 1, and m + n = (m+n), n neutrosophc structures developed n algebra, geometry, topology etc. The most developed felds of Netrosophc theory are Neutrosophc Set, Neutrosophc ogc, Neutrosophc Probablty, and Neutrosophc Statstcs - that started n 1995, and recently Neutrosophc Precalculus and Neutrosophc Calculus, together wth ther applcatons n practce. Neutrosophc Set and Neutrosophc ogc are generalzatons of the fuzzy set and respectvely fuzzy logc (especally of ntutonstc fuzzy set and respectvely ntutonstc fuzzy logc). n neutrosophc logc a proposton has a degree of truth (T), a degree of ndetermnacy (), and a degree of falsty (F), where T,, F are standard or nonstandard subsets of ] - 0, 1 + [. Mult-objectve lnear programmng problem (MOPP) a promnent tool for solvng many real decson-makng problems lke game theory, nventory problems, agrculture based management systems, fnancal and corporate plannng, producton plannng, marketng and meda selecton, unversty plannng and student admsson, health care and hosptal plannng, ar force mantenance unts, bank branches etc. 78

3 Neutrosophc Operatonal Research Volume Our objectve n ths chapter s to propose an algorthm to the soluton of neutrosophc mult-objectve programmng problem (NMOPP) wth the help of the frst order Taylor s theorem. Thus, neutrosophc mult-objectve lnear programmng problem s reduced to an equvalent mult-objectve lnear programmng problem. An algorthm s proposed to determne a global optmum to the problem n a fnte number of steps. The feasble regon s a bounded set. n the proposed approach, we have attempted to reduce computatonal complety n the soluton of (NMOPP). The proposed algorthm s appled to suppler selecton problem. The rest of ths chapter s organzed as follows. Secton 2 gves bref some prelmnares. Secton 3 descrbes the formaton of the problem. Secton 4 presents the mplementaton and valdaton of the algorthm wth practcal applcaton. Fnally, secton 6 presents the concluson and proposals for future work. 2 Some prelmnares Defnton 1. [1] A trangular fuzzy number J s a contnuous fuzzy subset from the real lne R whose trangular membershp functon J s defned by a contnuous mappng from R to the closed nterval [0,1], where (6) J 0 for all J, a J 1, (7) J s strctly ncreasng on J a m J (8) J 1 for J m, J 1,, (9) J s strctly decreasng on J m, a J (10) J 0 for all J Ths wll be elcted by: J a 2,. 2, J J J a1, a1 J m, m a 1 a2 J J, m J a2, a2 m 0, otherwse. (18) 79

4 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou Fg. 3: Membershp Functon of Fuzzy Number J. where m s a gven value a 1 & a 2 denotng the lower and upper bounds. Sometmes, t s more convenent to use the notaton eplctly hghlghtng the membershp functon parameters. n ths case, we obtan J a a J ; 1,, 2 Ma Mn,,0 m a1 a2 m 1 2 J a m a (19) n what follows, the defnton of the α-level set or α-cut of the fuzzy number J s ntroduced. Defnton 2. [1] et X = { 1, 2,, n } be a fed non-empty unverse. An ntutonstc fuzzy set FS A n X s defned as A,, X (20) A A whch s characterzed by a membershp functon : X 0,1 and a nonmembershp functon A : X 0,1 wth the condton 0 A A 1 for all X where A and A represent,respectvely, the degree of membershp and non-membershp of the element to the set A. n addton, for each FS A n X, 1 for all X s called the degree of hestaton A A A of the element to the set A. Especally, f 0,, then the FS A s degraded to a fuzzy set. A A 80

5 Neutrosophc Operatonal Research Volume Defnton 3. [4] The α-level set of the fuzzy parameters J n problem (1) s defned as the ordnary set J eceeds the level, α, α 0,1, where: J J R J J for whch the degree of membershp functon (21) For certan values j to be n the unt nterval. Defnton 4. [10] et X be a space of ponts (objects) and X. A neutrosophc set A n X s defned by a truth-membershp functon (), an ndetermnacy-membershp functon () and a falsty-membershp functon F(). t has been shown n fgure 2. (), () and F() are real standard or real nonstandard subsets of ]0,1+[. That s T A ():X ]0,1+[, A ():X ]0,1+[ and F A ():X ]0,1+[. There s not restrcton on the sum of (), () and F(), so 0 supt A () sup A () F A () 3+. n the followng, we adopt the notatons μ(), σ A () and v A () nstead of T A (), A () and F A (), respectvely. Also, we wrte SVN numbers nstead of sngle valued neutrosophc numbers. Defnton 5. [10] et X be a unverse of dscourse. A sngle valued neutrosophc set A over X s an object havng the form A={, μ A (), σ A (),v A () : X}, where μ A ():X [0,1], σ A ():X [0,1] and v A ():X [0,1] wth 0 μ A ()+ σ A ()+v A () 3 for all X. The ntervals μ(), σ A () and v A () denote the truthmembershp degree, the ndetermnacy-membershp degree and the falsty membershp degree of to A, respectvely. For convenence, a SVN number s denoted by A=(a,b,c), where a,b,c [0,1] and a+b+c 3. Defnton 6 et J be a neutrosophc trangular number n the set of real numbers R, then ts truth-membershp functon s defned as J a1, a1 J a2, a2 a 1 a2 J TJ J, a2 J a3, a3 a2 0, otherwse. (22) ts ndetermnacy-membershp functon s defned as 81

6 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou J b1, b1 J b2, b2 b 1 b2 J J J, b2 J b3, b3 b2 0, otherwse. (23) and ts falsty-membershp functon s defned as J c1, c1 J c2, c2 c 1 c2 J FJ J, c2 J c3, c3 c2 1, otherwse. (24) Fg. 4: Neutrosophcaton process [11] 3 Formaton of The Problem The mult-objectve lnear programmng problem and the mult- objectve neutrosophc lnear programmng problem are descrbed n ths secton. 82

7 Neutrosophc Operatonal Research Volume A. Mult-objectve Programmng Problem (MOPP) [6]: n ths chapter, the general mathematcal model of the MOPP s as follows mn/ ma,...,,,...,,...,,..., z 1 1 n z 2 1 n z p 1 n (8) subject to S, 0 n S R AX b, X 0. B. Neutrosophc Mult-objectve Programmng Problem (NMOPP) (25) f an mprecse aspraton level s ntroduced to each of the objectves of MOPP, then these neutrosophc objectves are termed as neutrosophc goals. et z z, z denote the mprecse lower and upper bounds respectvely for the th neutrosophc objectve functon. For mamzng objectve functon, the truth membershp, ndetermnacy membershp, falsty membershp functons can be epressed as follows: 1, f z z, z z z, f z z z, z z 0, f z z 1, f z z, z z z, f z z z, z z 0, f z z 0, f z z, z z z, f z z z, z z 1, f z z (26) (27) (28) For mnmzng objectve functon, the truth membershp, ndetermnacy membershp, falsty membershp functons can be epressed as follows: 83

8 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou 1, f z z, z z z, f z z z, z z 0, f z z 1, f z z, z z z, f z z z, z z 0, f z z 0, f z z, z z z, f z z z. z z 1, f z z (29) (30) (31) 4 Algorthm for Neutrosophc Mult-Objectve Programmng Problem The computatonal procedure and proposed algorthm of presented model s gven as follows: Step 1. Determne 1, 2,..., n that s used to mamze or mnmze the th truth membershp functon X, the ndetermnacy membershp X, and the falsty membershp functons X and n s the number of varables.. =1,2,..,p Step 2. Transform the truth membershp, ndetermnacy membershp, falsty membershp functons by usng frst-order Taylor polynomal seres n j j j 1 j (32) n j j j 1 (33) j 84

9 Neutrosophc Operatonal Research Volume n j j j 1 j (34) Step 3. Fnd satsfactory 1, 2,..., n by solvng the reduced problem to a sngle objectve for the truth membershp, ndetermnacy membershp, falsty membershp functons respectvely. p p 1 j1 q p 1 j1 h p 1 j1 n j j n j j n j j j j j (35) Thus neutrosophc multobjectve lnear programmng problem s converted nto a new mathematcal model and s gven below: Mamze or Mnmze p() Mamze or Mnmze q() Mamze or Mnmze h(), where X, X and X calculate usng equatons (10), (11), and (12) or equatons (13), (14), and (15) accordng to type functons mamum or mnmum respectvely. 4.1 llustratve Eample A mult-crtera suppler selecton s selected from [2]. For supplyng a new product to a market assume that three supplers should be managed. The purchasng crtera are net prce, qualty and servce. The capacty constrants of supplers are also consdered. t s assumed that the nput data from supplers performance on these crtera are not known precsely. The neutrosophc values of ther cost, qualty and servce level are presented n Table 1. The mult-objectve lnear formulaton of numercal eample s presented as mn z 1, ma z 2, z 3 : 85

10 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou mn z 5 7 4, ma z , ma z , st..: , , 450, 450, 0, 1,2,3. Table 1: Supplers quanttatve nformaton Z1 Cost Z2Qualty (%) Z3 Servce (%) Suppler Suppler Capacty Suppler The truth membershp, ndetermnacy membershp, falsty membershp functons were consdered to be neutrosophc trangular. When they depend on three scalar parameters (a1, m, a2). z 1 depends on neutrosophc aspraton levels (3550,4225,4900), when z 2 depends on neutrosophc aspraton levels (660,681.5,702.5), and z3 depends on neutrosophc aspraton levels (657.5,678.75,700). The truth membershp functons of the goals are obtaned as follows: 0, f z1 3550, 4225 z1, f 3550 z1 4225, z z1, f 4225 z1 4900, , f z , f z , z , f z , z 2 z 2 660, f 660 z , , f z

11 Neutrosophc Operatonal Research Volume f z 0, f z 3 700, z , f z 3 700, z 3 z , f z , , f z ma mn,, z 2 mn(ma(, ,1)) z 3 mn(ma(, ,1)) Then 350,0,450, 0,450,350, 400,0, The truth membershp functons are transformed by usng frst-order Taylor polynomal seres 350, 0, ,0, ,0, ,0, µ 1 () = n the smlar way, we get 1 µ 2 () = µ 3 () = The p() s p

12 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou P() = st..: 800, , 450, 450, 0, 1,2,3. The lnear programmng software NGO 15.0 s used to solve ths problem. The problem s solved and the optmal soluton for the truth membershp model s obtaned s as follows: ( 1, 2, 3 ) = (350, 0,450) z 1 =3550, z 2 =662. 5, z 3 = The truth membershp values are 1 1, , The truth membershp functon values show that both goals z 1, z 3 and z 2 are satsfed wth 100%, 11.63% and 89.4% respectvely for the obtaned soluton whch s 1=350; 2= 0, 3=450. n the smlar way, we get X, q(). Consequently, we get the optmal soluton for the ndetermnacy membershp model s obtaned s as follows: ( 1, 2, 3 )=(350,0,450) z 1 =3550, z 2 =662.5, z 3 =697.5 and the ndetermnacy membershp values are 1 1, , The ndetermnacy membershp functon values show that both goals z 1, z 3 and z 2 are satsfed wth 100%, 11.63% and 89.4% respectvely for the obtaned soluton whch s 1=350; 2= 0, 3=450. n the smlar way, we get X and h() Consequently, we get the optmal soluton for the falsty membershp model s obtaned s as follows: ( 1, 2, 3 )=(350,0,450) z 1 =3550, z 2 =662.5, z 3 =697.5 and the falsty membershp values are 1 0, , The falsty membershp functon values show that both goals z 1, z 3 and z 2 are satsfed wth 0%, 88.37% and 10.6% respectvely for the obtaned soluton whch s 1=350; 2= 0, 3= Conclusons and Future Work n ths chapter, we have proposed a soluton to Neutrosophc Multobjectve Programmng Problem (NMOPP). The truth membershp, ndetermnacy membershp, falsty membershp functons assocated wth each objectve of the problem are transformed by usng the frst order Taylor polynomal seres. The neutrosophc mult-objectve programmng problem s 88

13 Neutrosophc Operatonal Research Volume reduced to an equvalent multobjectve programmng problem by the proposed method. The soluton obtaned from ths method s very near to the soluton of MOPP. Hence ths method gves a more accurate soluton as compared wth other methods. Therefore, the complety n solvng NMOPP has reduced to easy computaton. n the future studes, the proposed algorthm can be solved by metaheurstc algorthms. Acknowledgement The authors would lke to thank the anonymous referees for ther constructve comments and suggestons that have helped a lot to come up wth ths mproved form of the chapter. References [1] El-Hefenawy, N., Metwally, M. A., Ahmed, Z. M., & El-Henawy,. M. A Revew on the Applcatons of Neutrosophc Sets. Journal of Computatonal and Theoretcal Nanoscence, 13(1), (2016), pp [2] X. Xu, Y. e, and W. Da. ntutonstc fuzzy nteger programmng based on mproved partcle swarm optmzaton. n: Journal of Computer Applcatons, vol. 9, 2008, p [3] A. Yücel, and Al Fuat Güner. A weghted addtve fuzzy programmng approach for mult-crtera suppler selecton. n: Epert Systems wth Applcatons, 38, no. 5 (2011): [4] Mohamed, Ma, et al. "Neutrosophc nteger Programmng Problem." Neutrosophc Sets & Systems 15 (2017). [5]. Y,. We-mn, and X. Xao-la. ntutonstc Fuzzy Blevel Programmng by Partcle Swarm Optmzaton. n: Computatonal ntellgence and ndustral Applcaton, PACA 08. Pacfc-Asa Workshop on, EEE, 2008, pp [6] Abdel-Baset, Mohamed, and brahm M. Hezam. An mproved Flower Pollnaton Algorthm for Ratos Optmzaton Problems. n: Appled Mathematcs & nformaton Scences etters An nternatonal Journal, 3, no. 2 (2015): [7] R. rene Hepzbah, R and Vdhya. ntutonstc Fuzzy Mult-Objectve near Programmng Problem(FMOPP) usng Taylor Seres. n: nternatonal Journal of Scentfc and Engneerng Research (JSER), Vol. 3, ssue 6, June [8] A. Amd, S. H. Ghodsypour, and Ch O Bren. Fuzzy multobjectve lnear model for suppler selecton n a supply chan. n: nternatonal Journal of Producton Economcs 104, no. 2 (2006): [9] D. Pntu, Tapan K. R. Mult-objectve non-lnear programmng problem based on Neutrosophc Optmzaton Technque and ts applcaton n Rser Desgn Problem. n: Neutrosophc Sets and Systems, Vol. 9, 2015: [10] Abdel-Basset, M., Mohamed, M. & Sangaah, A.K. J Ambent ntell Human Comput (2017). DO: [11] S. Aggarwal, Ranjt B., and A. Q. Ansar. Neutrosophc modelng and control. n> Computer and Communcaton Technology (CCCT), 2010 nternatonal Conference on, pp EEE, [12] Abdel-Baset, M., Hezam,. M., & Smarandache, F. Neutrosophc Goal Programmng. n: Neutrosophc Sets & Systems, vol. 11 (2016). 89

14 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou [13] Smarandache, F. A nfyng Feld n ogcs: Neutrosophc ogc. Neutrosophy, Neutrosophc Set, Neutrosophc Probablty: Neutrosophc ogc. Neutrosophy, Neutrosophc Set, Neutrosophc Probablty. nfnte Study,