A Piecewise Linear Approximation Method to Solve Fuzzy Separable Quadratic Programming Problem

Transcription

1 Iteratioal Joural o Advaced Computer Research (ISSN (prit): ISSN (olie): ) Volume-3 Number- Issue-8 March-03 A Piecewise Liear Approximatio Method to Solve uzz Separable Quadratic Programmig Problem Shamal Se, Bia Bara Pal Departmet o Mathematics, Brahmaada Keshab Chadra College, Kolata 70008, Idia Departmet o Mathematics, Uiversit o Kalai, Kalai-7435, Idia Abstract This paper presets a piecewise liear approximatio method or solvig separable quadratic programmig problems b usig liear uzz goal programmig (GP) methodolog. I the proposed approach, the obectives are irst described uzzil b itroducig imprecise aspiratio level to each o them. The uzz goals are the characterized b their associated membership uctios or represetatio o goal achievemet i terms o membership values o uzz goals.i the model ormulatio o the problem, the deied membership uctios are irst trasormed ito membership goals b assigig the highest membership value (uit) ad itroducig uderad over-deviatioal variables to each o them. The, the membership goals i quadratic orm are trasormed ito liear goals b usig piecewise liear approximatio method. I the solutio process, miimizatio o uder- deviatioal variables i the goal achievemet uctio uder the misum GP solutio approach is cosidered. To illustrate the proposed approach a umerical example is solved. The model solutio is also compared with the solutio achieved b usig Talor series approximatio method. Kewords uzz Programmig, Goal Programmig, uzz Goal Programmig, Piecewise Liear Approximatio.. Itroductio Most o the real-world decisio problems are multiobective i ature ad the colict to each other regardig optimizatio o obectives. To resolve the colict, the goal programmig (GP) approach has bee itroduced b Chares ad Cooper [] i 96. The mai problem o usig GP is that a precise aspiratio level eed be assiged or each o the obectives. But, i a real-lie decisio situatio, it is diicult to set precise target values to obectives due to imprecise i ature o huma udgmets. To 9 overcome such a situatio, uzz programmig (P) approach has bee itroduced b Bellma ad Zadeh [] i 970. Zimmerma irst proposed the uzz liear programmig [3] approach i 978. I P, membership uctios are deied o the basis o assiged aspiratio levels ad tolerace rages deied or the uzz goals. But, it is diicult to deie tolerace rages i a highl sesitive decisio situatio. To overcome such diiculties, goal programmig approach i uzz eviromet has bee irst itroduced b Narashima [4] i 980. Thereater, GP has bee studied extesivel [5] b the active researchers ad has bee applied to dieret real lie problems [6, 7]. There are ma real-world decisio problems i dieret structural optimizatio areas. It is oud that obectives o most o the idustrial problems are oliear i ature. To solve such problems, dieret classical approaches have bee developed ad widel circulated i the literature. Oe o most widel used approaches is liear approximatio o a oliear uctio. Two promiet methods o approximatio o uctios are Talor Series approximatio method ad piecewise liear approximatio method. Talor series approximatio method is oe o the most widel used or liearizatio o quadratic obectives. Talor series approximatio method has bee used b Tosari [8], Pal ad Moitra i [9] to the problems with ractioal ad quadratic obectives. But, the are rough approximatios methods, ad roud-o errors ma occur i practical decisio situatios. Separable programmig is a special brach o oliear programmig where the obectives are expressed as the sum o separable uctios o sigle variable. Separable programmig was irst itroduced b Miller [0] i 963. Thereater, it has bee developed b Cox [], Keha et al. [], Li ad Che [3], Chag [4], Zhag ad Wag [5], Croxto et al. [6], ad other researchers. But, separable programmig approach i the area o GP is ot widel circulated i literature. Geerall,

2 Iteratioal Joural o Advaced Computer Research (ISSN (prit): ISSN (olie): ) Volume-3 Number- Issue-8 March-03 separable programmig problems are trasormed ito liear orm b usig piecewise liear approximatio method [7]. Sice piecewise liear approximatio approach approximates uctios i a piecewise maer, the error estimatio becomes less i compariso to the Talor series approximatio method. I this paper, the piecewise liear approximatio method are addressed to solve uzz multiobective quadratic programmig problem. I the model ormulatio o the problem, irst the deied uzz goals are characterized algebraicall b itroducig associated tolerace limits. The membership uctios are the coverted ito membership goals b taig maximum attaiable membership value (uit) as the target level ad itroducig uder-ad over-deviatioal variables to each o them. I the solutio process, the oliear membership goals i quadratic orm are approximated to liear goals b usig piecewise liear approximatio method. The misum GP methodolog is emploed to solve the problem. The potetial use o the approach is illustrated b a umerical example.. Problem ormulatio The geeric orm multiobective separable programmig problem ca be preseted as: Max Z Mi Z subect to i (X) (X) i (x ), (x ), c x b, i,,..., m,,..., K (K ),(K ),..., K l x u,,,..., () where (x ) represets uivariate quadratic uctio o (x ) x x ad deied as x,,...,k;,,..., ad where, R, Let b be the imprecise aspiratio level o the -th obective Z (X) (=,,.,K). The the uzz goals tae the orm as: 30 Z (X) ~ b, =,,.,K Z (X). b = (K +), (K +),., K where X is the vector o decisio variables, ~ ad. represet the uzziess o ad restrictios respectivel, i the sese o Zimmerma [3]. The uzz goals are characterized b their membership uctios. The membership uctio or (X) ~ b appear as [5]: i Z (X) b, Z (X) l i l Z (X) b, (X) b l 0 i Z (X) l,,,...,k () Agai, or tpe o restrictio, (X). b taes the orm i Z (X) b, u Z (X) i b Z (X) u, (X) u b 0 i Z (X) u, (K ), (K ),...,K. (3) 3. uzz goal programmig ormulatio I the GP model ormulatio o the problem, the deied membership uctios are coverted ito the membership goals b itroducig uder- ad overdeviatioal variables ad assigig the highest membership value ( uit) as the aspiratio level to each o them. The, the membership goals ca be preseted as: Z (X) l d d,,,...,k (4) b l u Z (X) ad d d u b (K ), (K ),...,K. (5) where d, d 0 represet the uder- ad overdeviatioal variables cocered with achievemet o the aspired level o the -th membership goal. I uzz goal programmig ormulatio, sice maximum value o the membership uctio has bee

3 Iteratioal Joural o Advaced Computer Research (ISSN (prit): ISSN (olie): ) Volume-3 Number- Issue-8 March-03 set as aspiratio level, ol uder-deviatio associated with the respective goals are to be miimized to achieve the desired solutio. The weighted GP model ca be preseted as: Miimize Z= K w d, ad satis the goal expressios i (4), (5) ad subect to the sstem costraits i (), where w ( 0) represets the weight o importace o uwated deviatioal variable associated with the th goal ad deied as [5]: w b u l b, i, i is deied as i () is deied as i (3) 4. Piecewise liearizatio o quadratic Goal The goals i (4) ad (5) ca be explicitl expressed as: [ b l ad u b [u (x ) l ] d respectivel. d (x )] d (K ), (K,,,..., K d ),...,K To liearize the quadratic uctio (x ), the grid poits (brea poits) or the variable x (,,..., ) are chose as a p (p 0,,..., p ). Itroducig ew variables p (p 0,,..., p ), x ca be expressed as: x p p a p p p where ( 0 ) a p u. p0 p p with o a l ad The, the piecewise approximated liear orm o the quadratic uctio (x ) (desigated as ) ca be expressed as: p (a ) (6) p0 p p Usig the relatio i (6), the executable liear GP model ca be preseted as: Miimize Z= K w d so as to satis b ad u l b [ [u l ] d ] d p d d, (K where (x ) (a ) subect to p 0 c p0 ( p i p p p a p p,,...,k, (,,..., ; p,,...,p ), (K ) b, i,,..., m i ). ),...,K At most two p ma be positive ad i two are positive, the must be cosecutive. To eorce the above coditio biar variable z (,,..., ; p 0,,..., p ) are to be p itroduced. The required restrictios appear as: z 0 p p p 0 z 0 z p- z p p, p {,,..., p } z (7) p To illustrate the proposed approach, a umerical example is solved i Sectio Numerical example The maximizatio problem o two separable quadratic obectives is stated as: Maximize (X) x x x Z 3

4 Iteratioal Joural o Advaced Computer Research (ISSN (prit): ISSN (olie): ) Volume-3 Number- Issue-8 March-03 x x Maximize Z (X) x x Subect to 5 x x (8) Usig the costraits i (8), the bouds o the variables ca be obtaied as x 3 x 4 (9) Evaluatig the best ad worst idividual solutio o the obectives, the uzz goals ca be preseted as: Z (X) ~ 0, Z (X) ~ 6 Lower toleraces o the obectives are 5 ad 4, respectivel. Now, membership uctios are obtaied as: X) (/ 5)[Z (x) 5] ( ( X) (/ )[Z (x) 4] Each o the obective uctios ca be expressed as the sum o separable uctios which are show i the Table. Table : Separable uctios associated with the obectives (x) x x (x ) x (x ) (x ) x x Itroducig uder-ad over-deviatioal variables, the membership goals ca be expressed as: [(x) (x ) 5] d d 5 [ (x) (x ) 4] d d (0) The membership goals are quadratic i ature. These uctios are to be reduced i liear orm. To approximate the uctios, the sets o grid poits or the variables x ad x are {,.5,,.5, 3} ad {,.5, 3, 3.5, 4}, respectivel. Usig the proposed procedure the executable GP model ca preseted as: Miimize Z= d d 5 so as to satis [ 5] d d 5 [ 4] d d subect to z 0 z0 z z z 3 z z3 4 z 3 0 z 0 z0 z z z 3 z z3 4 z where 0 (, ; p 0,,, 3, 4 ) ad z p p 4 5 (,;p 0,,,3) are biar variables ad satis the coditios preseted as: z z z z0 3 z0 z z z3 Usig the sotware LINGO (Versio 6.0) the problem is solved ad the obtaied solutio is ( x, x) (, 4) with (Z, Z) (0, 6) The, the resultig membership values are achieved as:, The results show that the solutio is achieved accordig to the eeds ad desires o the decisio maer i the imprecise decisio eviromet. Note: The uctio Z (X) ca be approximated b usig irst order Talor series approximatio method about the poit X (x, x,...,x ) as: 3

5 Iteratioal Joural o Advaced Computer Research (ISSN (prit): ISSN (olie): ) Volume-3 Number- Issue-8 March-03 Z (x, x,...,x ) Z (x x (x Z ), x (x,...,x, x x ),...,x Now usig irst order Talor series approximatio about the poit (3, ), the liear equivalet goals o the quadratic goals i (0) ca be preseted as: ( / 5)[8x 4x 7] d d ( / )[6x 4x 8] d d () The the executable GP model ca be expressed as: Miimize Z= d d 5 so as to satis the goal equatios i () ad subect to the set o costraits i (8). Usig the sotware LINGO (Versio 6.0) the problem is solved ad the solutio is obtaied as: ( x, x) (3, ) with (Z, Z) (0,) The compariso o the obective values, obtaied b usig the proposed approach ad irst order Talor series approximatio, is show i the igure. igure : Compariso o the obective values obtaied b two dieret approaches. The Comparisos show that the solutio achieved uder the proposed approach is superior over the Talor series approximatio approach. 6. Coclusio I this paper, piecewise liear approximatio method has bee emploed or solvig uzz separable quadratic programmig problem. Uder the proposed approach, the accurac o solutio ca easil be made ) where eeded b icreasig the umber o grid poits without ivolvig a computatioal complexit. The proposed method ca be exteded to solve liear ractioal programmig problems, quadratic ractioal programmig problems ad bilevel programmig problems with separable obective uctios as well as costraits, which ma be the problems i uture studies. Acowledgmet The authors are thaul to the Orgaizig Chair ICACC 03, ad the aomous Reeree or their valuable commets ad suggestios to improve the clarit ad qualit o presetatio o the paper. Reereces [] A. Chares, ad W. W. Cooper, Maagemet Models ad Idustrial Applicatios o Liear Programmig, Vol. I ad II, Wile, New Yor, 96. [] R. E. Bellma, ad L. A. Zadeh, Decisio- Maig i a uzz Eviromet, Maagemet Scieces, 7, B4 B64, 970. [3] H. -J. Zimmerma, uzz Programmig ad Liear Programmig with Several Obective uctios, uzz Sets ad Sstem,, 45 55, 978. [4] R. Narasimha, O uzz Goal Programmig Some Commets, Decisio Scieces,, , 980. [5] B. B. Pal, B. N. Moitra, ad U. Mauli, A Goal Programmig Procedure or uzz Multiobective Liear ractioal Programmig Problem, uzz Sets ad Sstems, 39, , 003. [6] A. Biswas, ad B. B. Pal, Applicatio o uzz Goal Programmig Techique to Lad Use Plaig i Agricultural Sstem, Omega, 33, , 005. [7] M. A. Parra, A. B. Terol ad M. V. R. Uria, A uzz Goal Programmig Approach to Portolio Selectio, Europea Joural o operatioal Research, 33, 87 97, 00. [8] M. D. Tosari, Talor Series Approach to uzz Multiobective Liear ractioal Programmig, Iormatio Scieces, 78, 89-04, 008. [9] B. B. Pal, ad B. N. Moitra,, A Goal Programmig Procedure or Solvig Problems with Multiple uzz Goals Usig Damic Programmig, Europea Joural o Operatioal Research, 44(3), ,

6 Iteratioal Joural o Advaced Computer Research (ISSN (prit): ISSN (olie): ) Volume-3 Number- Issue-8 March-03 [0] C. E. Miller, The simplex Method or Local Separable Programmig, Recet Advaces i Mathematical programmig (R.L. Groves ad P. Wole Eds.), 89-00, 963. [] M. G. Cox, A Algorithm or Approximatig Covex uctio b Meas o the irst Degree Splies, Computer Joural, 4, 7-75, 97. [] A. B. Keha,, I. R. Jr. De ariasr., ad G. L. Nemhauser Model or Represetig Piecewise Liear Cost uctios, Operatios Research Letters, 3, 44-48, 004. [3] C. C. Li, ad A. P. Che, Geeralizatio o Yag et al. s Method or uzz Programmig with Liear Membership uctios, uzz Sets ad Sstems, 3, , 00. [4] C. T. Chag, A Eiciet Liearizatio Approach or Mixed Iteger Problems, Europea Joural o Operatioal Research, 3, , 000. [5] H. Zhag ad S. Wag, Liearl Costraied Global Optimizatio via Piecewise-Liear Approximatio, Joural o Computatioal ad Applied Mathematics, 4, -0, 008. [6] K. L. Croxto, B. Gedro, ad T. L. Maati, A Compariso o Mixed Iteger Programmig Models or No Covex Piecewise Liear Cost Miimizatio Problem, Maagemet Sciece, 49 (9), 68-73, 003. [7] M. S. Bazaraa, H. D. Sherali ad C. M. Shett, Noliear Programmig Theor ad Algorithms, Joh Wile & Sos, Newor, 993. Shamal Se is a Assistat Proessor i the Departmet o Mathematics, B.K.C. College, Kolata. He received the M. Sc. Degree i Mathematics rom Uiversit o Kalai i 99. He has bee awarded PhD b the Uiversit o Kalai i April, 0. He has published his papers i dieret atioal ad iteratioal ourals. His research iterests pertai to dieret areas o multiobective decisio maig i the ield o operatio research icludig chace costraied programmig i the area o sot computig. Bia Bara Pal is a Proessor i the Departmet o Mathematics, Uiversit o Kalai, Idia. He received his MSc i Mathematics, Uiversit o Kalai i 979, ad the DIIT i computatioal Mathematics ad Computer programmig rom Idia Istitute o Techolog (IIT), Kharagpur, Idia i 980. He was awarded the PhD b the Uiversit o Kalai i 988. He has published a umber o research articles i dieret atioal ad iteratioal ourals icludig Elsevier Sciece. He has provided reviewer services to dieret iteratioal ourals ad iteratioal coereces. He is a editorial board member o the ourals: Iteratioal Joural o Applied Maagemet Sciece ad Iteratioal Joural o Data Aalsis Techiques ad Strategies, Idersciece.com. His research iterests cover dieret areas o sot-computig, pervasive computig, ad multiobective decisio maig (MODM) i iexact decisio eviromet. 34