Neutrosophic Goal Programming

Transcription

1 Neutrosophc Goal Programmng Excerpt from NEUTROSOPHIC OPERATIONAL RESEARCH Volume I. Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou. Foreword by John R. Edwards. Preface by the edtors. Brussels (Belgum): Pons p. ISBN Ibrahm M. Hezam 1 Mohamed Abdel-Baset 2 * Florentn Smarandache 3 1 Department of computer Faculty of Educaton Ibb Unversty Ibb cty Yemen. E-mal: Ibrahzam.math@gmal.com 2 Department of Operatons Research Faculty of Computers and Informatcs Zagazg Unversty Sharqyah Egypt. E-mal: analyst_mohamed@yahoo.com 3Math & Scence Department Unversty of New Mexco Gallup NM USA. E-mal: smarand@unm.edu Abstract In ths chapter the goal programmng n neutrosophc envronment s ntroduced. The degree of acceptance ndetermnacy and rejecton of objectves s consdered smultaneous. In the two proposed models to solve Neutrosophc Goal Programmng Problem (NGPP) our goal s to mnmze the sum of the devaton n the model (I) whle n the model (II) the neutrosophc goal programmng problem NGPP s transformed nto the crsp programmng model usng truth membershp ndetermnacy membershp and falsty membershp functons. Fnally the ndustral desgn problem s gven to llustrate the effcency of the proposed models. The obtaned results of Model (I) and Model (II) are compared wth other methods. Keywords Neutrosophc optmzaton; Goal programmng problem. 1 Introducton Goal programmng (GP) Models was orgnally ntroduced by Charnes and Cooper n early 1961 for a lnear model. Multple and conflctng goals can be used n goal programmng. Also GP allows the smultaneous soluton of a system of Complex objectves and the soluton of the problem requres the establshment among these multple objectves. In ths case the model must be solved n such a way that each of the objectves to be acheved. Therefore the sum of the devatons from the deal should be mnmzed n the objectve functon. It s mportant that measure devatons from the deal should have a sngle scale because devatons wth dfferent scales cannot be collected. However the target value assocated wth each goal could be neutrosophc n the real-world applcaton. In 1995 Smarandache [17] startng from phlosophy (when [8] 63

2 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou 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 from decson makng and control theory (makng a decson not makng or hestatng) from accepted/rejected/pendng etc. and guded by the fact that the law of excluded mddle dd not work any longer n the modern logcs. [12] 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 unty 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. In 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>. In the other hand Neutrosophc Theory studes the ndetermnacy labeled as I wth In = I for n 1 and mi + ni = (m+n)i n neutrosophc structures developed n algebra geometry topology etc. The most developed felds of Netrosophc theory are Neutrosophc Set Neutrosophc Logc 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 Logc are generalzatons of the fuzzy set and respectvely fuzzy logc (especally of ntutonstc fuzzy set and respectvely ntutonstc fuzzy logc). In neutrosophc logc a proposton has a degree of truth (T) a degree of ndetermnacy (I) and a degree of falsty (F) where TIF are standard or nonstandard subsets of ] [. 64

3 Neutrosophc Operatonal Research Volume I The mportant method for mult-objectve decson makng s goal programmng approaches n practcal decson makng n real lfe. In a standard GP formulaton goals and constrants are defned precsely but sometmes the system am and condtons nclude some vague and undetermned stuatons. In partcular expressng the decson maker s unclear target levels for the goals mathematcally and the need to optmze all goals at the same needs to complcated calculatons. The neutrosophc approach for goal programmng tres to solve ths knd of unclear dffcultes n ths chapter. The organzaton of the chapter s as follows. The next secton ntroduces a bref some prelmnares. Sectons 3 descrbe the formaton of the Problem and develop two models to neutrosophc goal programmng. Secton 4 presents an ndustral desgn problem s provded to demonstrate how the approach can be appled. Fnally conclusons are provded n secton 5. 2 Some Prelmnares Defnton 1. [17] A real 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 [01] where (1) J 0 for all J a J 1 (2) J s strctly ncreasng on J a m J (3) J 1 for J m J J 1 (4) J s strctly decreasng on J m a J (5) J 0 for all J J a 2. Ths wll be elcted by: 2 J a1 a1 J m m a 1 a2 J J J m J a2 a2 m 0 otherwse. (1) 65

4 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou Fg. 1: Membershp Functon of Fuzzy Number J. Where m s a gven value a 1 and a 2 denote the lower and upper bounds. Sometmes t s more convenent to use the notaton explctly hghlghtng the membershp functon parameters. In ths case we obtan J a a J ; 1 2 Max Mn 0 m a1 a2 m 1 2 J a m a (2) In what follows the defnton of the α-level set or α-cut of the fuzzy number J s ntroduced. Defnton 2. [1] Let X = {x 1 x 2 x n } be a fxed non-empty unverse an ntutonstc fuzzy set IFS A n X s defned as A x x x x X (3) A A whch s characterzed by a membershp functon : X 01 and a nonmembershp functon : X 01 wth the condton x x A A 0 A A 1 for all x X where A and A represent respectvely the degree of membershp and non-membershp of the element x to the set A. In addton for each IFS A n X x 1 x x for all x X s called the degree of hestaton A A A of the element x to the set A. Especally f x 0 then the IFS A s degraded to a fuzzy set. A 66

5 Neutrosophc Operatonal Research Volume I Defnton 3. [4] The α-level set of the fuzzy parameters J n problem (1) s defned as the ordnary set L J exceeds the level α α 01 where: for whch the degree of membershp functon L J J R J J (4) For certan values j to be n the unt nterval. Defnton 4. [10] Let X be a space of ponts (objects) and x X. A neutrosophc set A n X s defned by a truth-membershp functon (x) an ndetermnacy-membershp functon (x) and a falsty-membershp functon (x). It has been shown n fgure 2. (x) (x) and (x) are real standard or real nonstandard subsets of ]0 1+[. That s T A (x):x ]0 1+[ I A (x):x ]0 1+[ and F A (x):x ]0 1+[. There s not restrcton on the sum of (x) (x) and (x) so 0 supt A (x) supi A (x) F A (x) 3+. In the followng we adopt the notatons μ(x) σ A (x) and v A (x) nstead of T A (x) I A (x) and F A (x) respectvely. Also we wrte SVN numbers nstead of sngle valued neutrosophc numbers. Defnton 5. [10] Let X be a unverse of dscourse. A sngle valued neutrosophc set A over X s an object havng the form A={ x μ A (x) σ A (x)v A (x) :x X} where μ A (x):x [01] σ A (x):x [01] and v A (x):x [01] wth 0 μ A (x)+ σ A (x)+v A (x) 3 for all x X. The ntervals μ(x) σ A (x) and v A (x) denote the truthmembershp degree the ndetermnacy-membershp degree and the falsty membershp degree of x to A respectvely. For convenence a SVN number s denoted by A=(abc) where abc [01] and a+b+c 3. Defnton 6. Let J be a neutrosophc 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. (5) ts ndetermnacy-membershp functon s defned as 67

6 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou J b1 b1 J b2 b2 b 1 b2 J I J J b2 J b3 b3 b2 0 otherwse. (6) 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. (7) Fg. 2: Neutrosophcaton process [11] 68

7 Neutrosophc Operatonal Research Volume I 3 Neutrosophc Goal Programmng Problem Goal programmng can be wrtten as: T x x1 x 2 x n Fnd... To acheve: z t k (8) Subject to x X where t are scalars and represent the target achevement levels of the objectve functons that the decson maker wshes to attan provded X s feasble set of the constrants. The achevement functon of the (8) model s the followng: k 1 2 (9) Mn w n w p 1 Goal and constrants: z n p t k x X n p 0 n p 0 n p are negatve and postve devatons from t target. The NGPP can be wrtten as: So as to: T x x1 x 2 x n Fnd... Mnmze z wth target value t acceptance tolerance a ndetermnacy tolerance d rejecton tolerance c Subject to x X g j x b j j m x n wth truth-membershp ndetermnacy-membershp and falsty-membershp functons: 69

8 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou 1 f z t I z 1 t z f t z t a a 0 f z t ar I z 0 f z t z t f t z t d d z 1 t f t d z t a a d 0 f z t a 0 f z t I z t z f t z t C C 1 f z t C (10) (11) (12) Fg. 3: Truth-membershp ndetermnacy-membershp and falstymembershp functons for z. To maxmze the degree the accptance and ndetermnacy of NGP objectves and constrants also to mnmze the dgree of rejecton of NGP objectves and constrants Max z z k Max z z k (13) Mn z z k 70

9 Neutrosophc Operatonal Research Volume I Subject to 0 z z z z z z k z z k z z z z k z z z z k g j x b j j m x X x j 0 j n where z z z are truth membershp functon ndetermnacy z z z membershp functon falsty membershp functon of Neutrosophc decson set respectvely. The hghest degree of truth membershp functon s unty. So for the defned the truth membershp functon z the flexble membershp goals havng the aspred level unty can be presented as z z n 1 p 1 1 z For case of ndetermnacy (ndetermnacy membershp functon) t can be wrtten: z n 2 p z For case of rejecton (falsty membershp functon) t can be wrtten Here devatonal varables. z n 3 p 3 0 z n p n p n and p are under-devatonal and over- Our goals are maxmze the degree of the accptance and ndetermnacy of NGP objectves and constrants and mnmze the dgree of rejecton of NGP objectves and constrants. as: Model (I). The mnmzaton of the sum of the devaton can be formulated (14) Mn k k k 1 w 1 n 1 1 w 2 n 2 1 w 3 p 3 Subject to z z n k 71

10 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou z z n k z z p k z z k z z z z k z z z z k 0 z z z z z z k g j x b j j m n1 n 2 p k x X x j 0 j n On the other hand neutrosophc goal programmng NGP n Model (13) can be represented by crsp programmng model usng truth membershp ndetermnacy membershp and falsty membershp functons as: Max Max Mn (15) z z z z k z k z k z t k g j x b j j m x j 0 j n In model (15) the Max Max are equvalent to Mn 1 Mn 1 respectvely where (16) Mn Subject to z t a a d k z t k 72

11 Neutrosophc Operatonal Research Volume I g j x b j j m x j 0 j n If we take 1 1 v the model (16) becomes: Model (II). Mnmze v (17) Subject to z t a a d v k z t k g j x b j j m x j 0 j n The crsp model (17) s solved by usng any mathematcal programmng technque wth v as parameter to get optmal soluton of objectve functons. 4 Illustratve Example Ths ndustral applcaton selected from [15]. Let the Decson maker wants to remove about 98.5% bologcal oxygen demand (BOD) and the tolerances of acceptance ndetermnacy and rejecton on ths goal are and 0.3 respectvely. Also Decson maker wants to remove the sad amount of BODS 5 wthn 300 (thousand $) tolerances of acceptance ndetermnacy and rejecton (thousand $) respectvely. Then the neutrosophc goal programmng problem s: mn z x x x x 19.4x 16.8x x4 91.5x 120 mn z x x x x x x x x st..: x

12 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou Wth target 300 acceptance tolerance 200 ndetermnacy tolerance 100 and rejecton tolerance 300 for the frst objectve z 1. Also wth target acceptance tolerance 0.1 ndetermnacy tolerance 0.05 and rejecton tolerance 0.2 for the second objectve z 2. Where x s the percentage BOD5(to remove 5 days BOD) after each step. Then after four processes the remanng percentage of BOD5 wll be x = The am s to mnmze the remanng percentage of BOD5 wth mnmum annual cost as much as possble. The annual cost of BOD5 removal by varous treatments s prmary clarfer trcklng flter actvated sludge carbon adsorpton. z 1 represent the annual cost. Whle z 2 represent removed from the wastewater. The truth membershp ndetermnacy membershp falsty membershp functons were consdered to be neutrosophc trangular. The truth membershp functons of the goals are obtaned as follows: 1 f z1 300 I z z1 1 f 300 z f z f z I z z 2 1 f z f z The ndetermnacy membershp functons of the goals are gven: 0 f z1 300 z1 300 f 300 z1 400 I z1 z f 400 z f z f z z f z I z 2 z f z f z The falsty membershp functons of the goals are obtaned as follows: 74

13 Neutrosophc Operatonal Research Volume I 0 f z1 300 I z z1 f 300 z f z f z I z z 2 f z f z The software LINGO 15.0 s used to solve ths problem. Table (1) shows the comparson of the obtaned results among the proposed models and the others methods. Table 1: Comparson of optmal soluton based on dfferent methods: Methods z 1 z 2 x 1 x 2 x 3 x 4 FG 2 P 2 Ref[15] IFG 2 P 2 Ref[15] Model (I) Model (II) E It s to be noted that model (I) offers better solutons than other methods. 5 Conclusons and Future Work The man purpose of ths chapter was to ntroduce goal programmng n neutrosophc envronment. The degree of acceptance ndetermnacy and rejecton of objectves are consdered smultaneously. Two proposed models to solve neutrosophc goal programmng problem (NGPP) n the frst model our goal s to mnmze the sum of the devaton whle the second model neutrosophc goal programmng NGP s transformed nto crsp programmng model usng truth membershp ndetermnacy membershp and falsty membershp functons. Fnally a numercal experment s gven to llustrate the effcency of the proposed methods. Moreover the comparatve study has been held of the obtaned results and has been dscussed. In the future studes the proposed algorthm can be solved by metaheurstc algorthms. 75

14 Edtors: Prof. Florentn Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou References [1] O.M. Saad B.M.A. Hassan and I.M. Hzam Optmzng the underground water confned steady flow usng a fuzzy approach. In: Internatonal Journal of Mathematcal Archve (IJMA) ISSN vol [2] X. Xu Y. Le and W. Da Intutonstc fuzzy nteger programmng based on mproved partcle swarm optmzaton. In: Journal of Computer Appl. vol [3] A. Yücel and Al Fuat Güner. A weghted addtve fuzzy programmng approach for mult-crtera suppler selecton. In: Expert Systems wth Applcatons 38 no. 5 (2011): [4] I.M. Hzam O. A. Raouf M. M. Hadhoud: Solvng Fractonal Programmng Problems Usng Metaheurstc Algorthms Under Uncertanty. In: Internatonal Journal of Advanced Computng vol p [5] L. Y L. We-mn and X. Xao-la. Intutonstc Fuzzy Blevel Programmng by Partcle Swarm Optmzaton. In: Computatonal Intellgence and Industral Applcaton PACIIA 08. Pacfc-Asa Workshop on IEEE 2008 pp [6] Abdel-Baset Mohamed and Ibrahm M. Hezam. An Improved Flower Pollnaton Algorthm for Ratos Optmzaton Problems. In: Appled Mathematcs & Informaton Scences Letters An Internatonal Journal 3 no. 2 (2015): [7] R. Irene Hepzbah R and Vdhya Intutonstc Fuzzy Mult-Objectve lnear Programmng Problem(IFMOLPP) usng Taylor Seres. In: Internatonal Journal of Scentfc and Engneerng Research (IJSER) Vol. 3 Issue 6 June [8] A. Amd S. H. Ghodsypour and Ch. O Bren. Fuzzy multobjectve lnear model for suppler selecton n a supply chan. In: Internatonal 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. In: Neutrosophc Sets and Systems Vol : [10] Abdel-Basset M. Mohamed M. & Sangaah A.K. J Ambent Intell Human Comput (2017). DOI: [11] S. Aggarwal Ranjt B. and A. Q. Ansar. Neutrosophc modelng and control. In: Computer and Communcaton Technology (ICCCT) 2010 Internatonal Conference on pp IEEE [12] Mohamed Ma et al. "Neutrosophc Integer Programmng Problem." Neutrosophc Sets & Systems 15 (2017). [13] Dey S. & Roy T. K. "Intutonstc Fuzzy Goal Programmng Technque for Solvng Non-Lnear Mult-objectve Structural Problem." Journal of Fuzzy Set Valued Analyss 2015 no. 3 (2015): [14] Pramank P. & Roy T. K. An ntutonstc fuzzy goal programmng approach to vector optmzaton problem. Notes on Intutonstc Fuzzy Sets 11 no. 1 (2005): [15] Ghosh P. & Roy T. K. Intutonstc Fuzzy Goal Geometrc Programmng Problem (IF G2 P2) based on Geometrc Mean Method. In Internatonal Journal of Engneerng Research and Technology vol. 2 no. 11 (November-2013). ESRSA Publcatons [16] IM Hezam M Abdel-Baset F. Smarandache. Taylor Seres Approxmaton to Solve Neutrosophc Multobjectve Programmng Problem. In: Neutrosophc Sets and Systems Vol : [17] El-Hefenawy N. Metwally M. A. Ahmed Z. M. & El-Henawy I. M. A Revew on the Applcatons of Neutrosophc Sets. Journal of Computatonal and Theoretcal Nanoscence 13(1) (2016) pp