A Computatoal Procedure for solvg a No-Covex Mult-Obectve Quadratc Programmg uder Fuzz Evromet Shash Aggarwal * Departmet of Mathematcs Mrada House Uverst of Delh Delh-0007 Ida shash60@gmal.com Uda Sharma Departmet of Mathematcs Uverst of Delh Delh-0007 Ida udasharm88@ahoo.com The purpose of ths paper s to stud a o-covex fuzz mult-obectve uadratc programmg problem whch both the techologcal coeffcets ad resources are fuzz wth olear membershp fucto. A computatoal procedure to fd a fuzz effcet soluto of ths problem s developed. A umercal example s gve to llustrate the procedure. Kewords: No-covex Fuzz mult-obectve uadratc programmg olear membershp fucto Fuzz effcet soluto.. Itroducto Ths paper studes the problem of maxmzg a umber of uadratc obectve fuctos subect to lear ad boud costrats uder fuzz evromet: (NFMOQPP) Max Z t t x = cx + x Q x Max Z t t x = cx + x Q x where Max Z t t x = c x + x Q x A x k k k Subect to x X = {x R : c = k are -dmesoal cost vectors; b l x u x 0} Q = k are smmetrc postve sem-defte matrces; A s the m costrat fuzz matrx of techologcal coeffcets; b s the m-dmesoal fuzz resource vector -dmesoal fuzz vectors l ad u are lower ad upper bouds respectvel o -dmesoal decso vector x. * Correspodg Author.
Remark.. The problem (NFMOQPP) s a o-covex programmg problem because Q = k are assumed to be smmetrc postve sem-defte matrces ad our problem s maxmzato. I the lterature secto a greater part of the stud focused o covex programmg problem; as covex programmg local optma s global optma but case of o-covex programmg local optma ma ot be global optma. Orde (963) was the frst oe who cosdered the maxmzato of a o-cocave uadratc programmg. He derved the ecessar ad suffcet codtos for the maxmzato of o-cocave uadratc programmg. Rtter (966) exteded Orde s work ad gave a dfferet approach to fd the maxmzato of o-cocave uadratc programmg. Murt ad Kabad (987) dscussed that o-covex problem s a N-P Hard problem ad t s ver hard to get the global maxma. The have coferred that descet algorthms are ute practcal algorthms for dealg wth o-covex olear problem. Haso (990) used geeralzed vext to derve the ecessar ad suffcet codtos o-covex uadratc programmg for global mma. Ye (99) appled the affe scalg algorthm to fd the optmal soluto for the o-covex uadratc programmg. Burer ad Vadebussche (008) has developed a fte brach ad boud algorthm for o-covex uadratc programmg va sem defte relaxato. Che ad Burer (0) exteded the work of Burer ad Vadebussche to get the global optma of ocovex problem wth lear ad boud costrats. The purpose of ths paper s to develop a computatoal procedure for more geeral o-covex uadratc problem smlar to that of Che ad Burer (0) ad Orde (963). The have focused o the sgle obectve o-covex uadratc programmg problem. But here ths paper we have cocetrated o the problem (NFMOQPP) whch stead of sgle obectve we have take multple obectves whch are fuzzfed durg the procedure order to obta the bouds o them; the crsp lear ad boud costrats have bee replaced b fuzz lear ad boud costrats. I real world the cocept of decso-makg takes place a evromet whch the obectves ad costrats are ot kow precsel. Such stuatos ca be tackled effcetl wth the help of fuzz set theor. Fuzz sets were frst troduced b Zadeh (965). These sets were used to troduce the cocept of decso-makg a fuzz evromet b Bellma ad Zadeh (970). The have defed approprate aggregato of fuzz sets for the fuzz decso. Zmmerma (978) has used fuzz decso cocept the fuzz lear programmg for several obectves. Guu ad Wu (997) exteded the Zmmerma s approach to two-phase approach for solvg the mult-obectve lear programmg the fuzz evromet. Several authors have studed lear programmg the fuzz evromet ad appled to real world problems lke trasportato producto plag etc. A fuzz decso s based o the tersecto of membershp fuctos of the goals ad costrats. Most of the authors have used lear membershp fuctos as the are eas to tackle. Leberlg (98) has defed the hperbolc membershp fucto. L ad Lee (99) have defed the expoetal membershp fucto ad Yag Igzo ad Km (99) have defed pecewse olear membershp fucto. I our paper we have used trgoometrc membershp fucto terms of s for obectve fuctos ad some of the costrats ad used LINGO 9.0 to fd the fuzz effcet soluto of (NFMOQPP). The paper s orgazed as follows: Secto cossts of prelmares whch cota a useful defto ad some basc membershp fuctos of fuzz techologcal coeffcets ad fuzz resource vector; fuzz effcet soluto ad olear membershp fuctos are descrbed Secto 3; Secto 4 we have dscussed the soluto methodolog for solvg ocovex fuzz mult-obectve uadratc programmg problem; ad llustratve umercal example to expla the soluto methodolog s gve Secto 5.
. Prelmares I ths secto we gve a basc defto ad membershp fuctos correspodg to (NFMOQPP). Geeral form of No-covex Mult-obectve Quadratc Programmg problem s gve b t t (NMOQPP) Max Z x = cx + x Q x x = c x + x x t t Max Z Q Max Z t t x = c x + x Q x k k k where s the m Subect to x X = {x R : Ax b x 0} c = k are -dmesoal cost vectors; decso vector. Q = k are smmetrc postve sem-defte matrces; A costrat matrx of techologcal coeffcets; b s the m-dmesoal resource vector ad x s -dmesoal Defto.. A pot x X s sad to be Pareto optmal soluto of (NMOQPP) f there does ot exst a x X such that Z (x) Z (x ) = k ad Z r (x) > Z r ( x ) for at least oe r = k. () I the problem (NFMOQPP) A s the m costrat fuzz matrx of techologcal coeffcets; b s the m-dmesoal fuzz resource vector fuctos are gve as below: l ad u are -dmesoal fuzz vectors ad x s -dmesoal decso vector. Ther membershp. The membershp fucto of b : b b b p b p 0 b b p p () where R ad p >0 s the tolerace level of b for all = m.. The membershp fucto of the fuzz matrx A : where A R ad a a d a a +d d 0 a +d d > 0 s the tolerace level of a for =.m ad =... (3) 3
3. The membershp fucto of u : u u t u t 0 u u u t t (4) where R ad t >0 s the tolerace level of u for all =. 4. The membershp fucto of l : l l - l r l r r 0 l r l (5) where R ad r >0 s the tolerace level of l for all =. 3. Fuzz Effcet Soluto Werers (987) has gve a defto of fuzz effcet soluto for fuzz mult-obectve lear programmg problem. As our problem (NFMOQPP) the costrat set X s descrbed exclusvel b fuzz costrats the varg degree of feasblt should be take to accout b cosderato of effcet solutos for addtoal depedeces betwee the dvdual goal values ad the degree of membershp to the rego of feasble solutos ca arse. That meas we wat to emphasze ot ol the achevemet of maxmum value of obectve fuctos but also the hghest membershp grade of fuzz costrats a flexble rego. I order to take care of fuzz cocept we wll exted the defto of Werers to (NFMOQPP). Let the costrat sets of (NFMOQPP) be deoted b C (x) = { x R : A x b } =. m B(x) { x R : x u }... B (x) { x R : x l }... Defe the trgoometrc membershp fucto of th costrat C (x) as: 0 b a x a x a b dx p a d p C (x) s x ( )x b b ( )x a d p (6) where x R ad d > 0 ad p > 0 are respectvel the tolerace level of the techologcal coeffcets ad resources for = m ad =. 4
Ths trgoometrc membershp fucto has the followg propertes:. ( C (x)) s a olear ad mootocall decreasg fucto.. 0 ( C (x)) x R. 3. ( C (x)) s a cocave fucto o the set x R : a x b. We defe the lear membershp fucto of the costrat B (x) = as: where x R x u u t x ( B (x)) u x u t t ad t >0 s the tolerace level of u. Defe the lear membershp fucto of the costrat 0 x u + t l x B (x) = as : x l +r ( B (x)) l - r x l r 0 l - r x (8) (7) where x R ad r >0 s the tolerace level of l. Defto 3.. The pot x X s sad to be fuzz effcet soluto for the (NFMOQPP) f there does ot exst a x X such that Z x Z (x ) =...k ad C x C (x ) =...m ad B x B (x ) =... ad B x B (x ) =... ad Z x Z (x ) for at least oe =...k or C x C (x ) for at least oe =...m or B x B (x ) for at least oe =... or B x B (x ) for at least oe =... As metoed Werers (987) smlarl here ths defto takes to accout that for a fuzz effcet soluto a mprovemet cocerg a obectve fucto ca ol be reached ether at the expese of a addtoal obectve fucto or at the expese of the membershp to the costrats. I fact we ca easl see that the fuzz effcec defed above comprses the classcal effcec as specal case f each of the ( C (x)) ( B (x)) ad ( B (x)) s for each x C (x) x B (x) ad x B (x) respectvel. 4. Procedure to fd a Fuzz Effcet Soluto Our obectve fuctos are crsp ad costrats are fuzz ature (NFMOQPP). To defuzzfcate the problem we wll frst fuzzf the obectve fuctos whch ca be doe b solvg the followg four uadratc programmg problems 5
(NQPP ) Z = Max Z (x) = c t x + x t Q x = k Subect to x a x b = m u =. l x =... x 0. (NQPP ) Z = Max Z (x) = c t x + x t Q x = k Subect to a d x b = m x u =. l - r x =... x 0. (NQPP 3 ) Z 3 = Max Z (x) = c t x + x t Q x = k Subect to a d x b + p = m x u t =. l - r x =... x 0. (NQPP 4 ) Z 4 = Max Z (x) = c t x + x t Q x = k Subect to ax b + p = m x u t =. l x =... x 0. We have ow four o-covex uadratc problems correspodg to th ( =..k) obectve fucto. The ca be solved b usg LINGO 9.0 to get the asprato level for the th obectve fucto. Now we take the best ad worst value of the optmal solutos of the four o-covex uadratc programmg problems. Let Z L = M (Z Z Z 3 Z 4 ) ad Z U = Max (Z Z Z 3 Z 4 ) = k. (9) Now takg the terval [Z L Z U ] as the tolerace level terval for the th obectve (= k) (NFMOQPP) we defe the trgoometrc membershp fucto of the th obectve (= k) as: 6
0 Z x L L U Z x Z Z (x) s Z Z x Z U l Z - Z Z x Z Z L U (0) Ths trgoometrc membershp fucto has the followg propertes:. ( Z (x)) s a olear ad mootocall creasg fucto. L U. 0 ( Z (x)) Z Z. 3. ( Z (x)) s cocave fucto Z. L Now b usg max-m fuzz decso makg approach gve b Bellma ad Zadeh (970) we have x M Z (x) C (x) ( B (x)) ( B (x)) () D The the optmal fuzz decso s a soluto of the problem Ma x D (x) = x 0 Max ( M ( ( Z (x)) ( C (x)) ( B (x)) ( B (x)))). () x 0 The problem () s euvalet to the followg problem as dscussed b Zmmerma (978) (NPP) Max Subect to ( Z (x)) =...k ( C (x)) =...m ( B (x)) =... ( B (x)) =.... 0 x 0. Now the above problem s a olear covex programmg problem whch ca be solved b LINGO 9.0. Let ( x optmal soluto of the problem (NPP). ) be the Now b two phase method proposed b Guu ad Wu (997) we wll solve the followg olear problem: (NPP) Max k m + + Subect to Z x = k C x = m ( B (x)) ( B (x)) x 0. =. We solve the above problem b LINGO 9.0. Let ( x ) where (.. k ) = (.. m ) = ad be the optmal soluto of (NPP). Now we wll prove that ths optmal soluto ( x ) gves the fuzz effcet soluto of (NFMOQPP). 7
Theorem 4.. Let ( x ) where (... k ) (... m ) = ad (NFMOQPP). be the optmal soluto of the problem (NPP) the x s the fuzz effcet soluto of the problem Proof: Let f possble x be ot a fuzz effcet soluto of (NFMOQPP) the there exst a X such that Z Z (x ) =...k ad C C (x ) =...m ad B B (x ) =... ad B B (x ) =... ad (3) Z > Z (x ) for at least oe =...k or C > C (x ) for at least oe =...m or B > B (x ) for at least oe =... or B > B (x ) for at least oe =... As ( Z (.)) s the creasg fucto Z () Z ( x ) ( Z ( )) ( Z (x )) for all = k. Sce ( x ) s the optmal soluto ad the coeffcets the obectve fucto are postve problem (NPP) So we have = Z x = k = C x = m = B (x ) = = B (x ) = Now choosg = = Z = k C = m = B ( ) = = B ( ) = As ( x ) s the feasble soluto of (NPP) so usg eualtes (3) ad propert of ( Z (x)) we get Z Z x = k C C x = m ( B ( )) ( B (x )) ( B ( )) ( B (x )) =. So we ca easl see that ( ) where (... ) = ad k s the feasble soluto of (NPP). m (... ) As ( ) ad (x ) are solutos of the problem (NPP) ad usg (3) we get 8
k m + + k m = (Z ()) + ( C ( )) + B ( ) + B ( ) k m > C B B (Z (x )) + ( (x )) + x + x = k m + + Ths s a cotradcto as (x ) s optmal soluto of the problem (NPP). So x s the fuzz effcet soluto of (NFMOQPP). 4. Computatoal Procedure The deas dscussed above for fdg fuzz effcet soluto of (NFMOQPP) ca be summarzed the form of algorthm as gve below. Step : Costruct the four problems of the form (NQPP ) (= 3 4) correspodg to th (= k) obectve. Step : Solve them to get the asprato level for the th (=. k) obectve. Step 3: Determe the asprato level of the th ( = k) obectve b u 3 4 l 3 4 Z = Max Z Z Z Z ad Z = M Z Z Z Z =...k. Step 4: Costruct the trgoometrc membershp fuctos of the form (6) ad (0) for the costrats ad the obectves respectvel. Also Step 5: lear membershp fuctos of the form (7) ad (8) respectvel for upper ad lower values of x. Costruct the problem (NPP) ad solve t. Let ( x ) be the soluto ad be the optmal value. Step 6: Costruct the problem (NPP) ad solve t. Let ( x ) where (.. k ) = (.. m ) = ad be the optmal soluto. The x s the fuzz effcet soluto of problem (NFMOQPP). 5. Illustratve Example (NFMOQPP) Max Z (x) x + x + x + x Max Z x 4x + 7x + x + 3x Subect to x + x 0 x + 3x 5 x 9 x 8 x x 0. 9
Here A= (a ) = 3 (d ) = (a + d ) = 3 5 b = 0 5 5 5 p = 0 (b+ p) = 35 u = 9 3 8 t = u+ t = 0 l = r = l - r = Step: Computatoal Procedure: Costructo of the four problems of the form (NQPP ) (= 3 4) correspodg to th (= ) obectve as follows: (NQPP ) Z = Max (NQPP ) Z = Max (NQPP 3 ) 3 Z = Max (NQPP 4 ) 4 Z = Max x + x + x + x x + x 0 Subect to x + 3x 5 x 9 x 8 x x 0. x + x + x + x x + x 0 Subect to 3x + 5x 5 x 9 x 8 x x 0. x + x + x + x x + x 5 Subect to 3x + 5x 35 x x 0 x x 0 x + x + x + x x + x 5 Subect to x + 3x 35 x x 0 x x 0. (NQPP ) Z = Max (NQPP ) Z = Max (NQPP 3 ) 3 Z = Max (NQPP 4 ) 4 Z = Max 4x + 7x + x + 3x x + x 0 Subect to x + 3x 5 x 9 x 8 x x 0. 4x + 7x + x + 3x x + x 0 Subect to 3x + 5x 5 x 9 x 8 x x 0. 4x + 7x + x + 3x x + x 5 Subect to 3x + 5x 35 x x 0 x x 0 4x + 7x + x + 3x x + x 5 Subect to x + 3x 35 x x 0 x x 0. 0
Step : Now b solvg the above uadratc programmg problems usg LINGO 9.0 we get 3 4 Z = ( Z Z Z Z ) = (8 4 96.7 ad 8.75) 3 4 Z = ( Z Z Z Z ) = ( 8 73.68 ad 39.5) Step 3: Now usg (9) for = we get the terval [Z L Z U ] as the asprato level terval for the th obectve (= ) (NFMOQPP) as L U L U [ Z Z ] = [4 8.75] ad [ Z Z ] = [8 39.5] Step 4: Costructo of the membershp fuctos of the form (6) to (8) ad (0) for the costrats ad obectves as follows: 0 0 x + x 0 - x - x ( C (x)) s x + x 0 x +x +5 x + x + 5 0 x +x + 5 0 5 x + 3x 5 - x - 3x ( C (x)) s x + 3x 5 3x + 5x +0 x + x + 0 5 3x + 5x +0 x 9 ( B (x)) x 3 9 x 0 x x 8 ( B (x)) 0 x 8 x 0 0 x 0 0 x ( B (x)) (x ) x x 0 x ( B (x)) (x ) x x
(C (x)) (Z (x)) 0 Z x 4 Z x 4 Z (x) s 4 Z x 8.75 86.75 Z x 8.75 0 Z x 8 Z x 8 Z (x) s 8 Z x 39.5 30.5 Z x 39.5 Fgure Graph of (C (x)) Fgure Graph of (Z (x)) 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0. 0. 0. 0. 0 0 4 6 8 0 4 6 8 0 x +x 0 0 00 00 300 400 500 600 700 800 Z (x) = x + *x + x *x + *x *x From the graph draw fgure correspodg to the membershp fucto μ(c (x)) we ca observe that. ( C (x)) s a olear ad mootocall decreasg fucto.. 0 ( C (x)) x R. 3. ( C (x)) s a cocave fucto o the set x R : x x 0. Step 5: Also from the graph draw fgure correspodg to the membershp fucto μ(z (x)) we ca observe that. ( Z (x)) s a olear ad mootoc creasg fucto.. 0 ( Z (x)) x R. 3. ( Z (x)) s cocave fucto o the terval 4. Smlar propertes ca be observed for all other membershp fuctos wth the help of the graphs. Now the problem (NPP) becomes
(NPP) Max Subect to x + x + x + x - 4 s 86.75 4x + 7x + x + 3x 8 s 30.5 0 - x- x s x + x + 5 5 - x - 3x s x + x + 0 x 3 0 x (x ) (x ) 0 x x 0. We solve the above olear problem b LINGO 9.0. We get the optmal soluto as ( = 0.33004 x =.3300 x = 5.804374). Step 6: Now costruct the problem (NPP) as follows: (NPP) Max + + + + Subect to x + x + x + x - 4 s 86.75 0.33004 4x + 7x + x + 3x 8 s 0.33004 30.5 0 - x- x s 0.33004 x + x + 5 5 - x - 3x s 0.33004 x + x + 0 x 3 0 x 0.33004 0.33004 (x- ) 0.33004 3
(x- ) 0.33004 0.33004 x x 0. We solve the above problem (NPP) b LINGO 9.0 ad get the optmal values as ( = 0.33004 = 0.340065 = 0.36496 = 0.33004 = = = 0.33004 = x =.3300 x = 5.804373 ). Hece the fuzz effcet soluto of the problem (NFMOQPP) s x =.33004 ad x = 5.804373. The correspodg values of the obectves are Z (x ) = 8.09806 ad Z x = 50.569993. Ackowledgemets We are debted for ver helpful dscusso to Prof. Davder Bhata (Rtd.) Departmet of Operatoal Research Facult of Mathematcal Sceces Uverst of Delh Delh 0007 Ida. The secod author s thakful to the COUNCIL OF SCIENTIFIC AND INDUSTRIAL RESEARCH (CSIR) NEW DELHI INDIA for the facal support (09/045(53)/0-EMR-I). Refereces Bector C.R. ad S. Chadra (005). Fuzz Mathematcal programmg ad fuzz matrx games book. Sprger. Bellma R.E. ad L.A. Zadeh (970). Decso- makg a fuzz evromet. Maagemet Scece7(B)No.44-64. Burer S. ad D. Vadebussche (008). A fte brach-ad-boud algorthm for o-covex uadratc programmg va semdefte relaxatos. Mathematcal Programmg. Ser. A 3 59-8. Che J. ad S. Burer (0). Globall solvg o-covex uadratc programmg problems va completel postve programmg. Mathematcs ad Computer Scece Dvso Argoe Natoal Laborator Argoe Illos. GuuS.M. ad Y.K.Wu (997). Weghted coeffcets two-phase approach for solvg the multple obectve programmg problems. Fuzz Sets ad Sstems 85 45-48. Gasmov R. N.(00).Augmeted Lagraga Dualt ad odfferetable optmzato methods o-covex programmg. Joural of Global Optmzato 4 87-03. Haso M.A. (990). No-covex uadratc programmg. FUS Techcal report umber M-80 Departmet of statstcs Florda State Uverst. Leberlg H. (98). O fdg compromse solutos multcrtera problems usg the fuzz m-operator. Fuzz sets ad sstems 6 05-8. L R. J. ad E. Stale Lee (99). A expoetal membershp fucto for fuzz multple obectve lear programmg. Computers ad mathematcs wth applcatos No.- 55-60. Murt K.G. ad S.N. Kabad (987). Some NP-complete problems uadratc ad olear programmg. Mathematcal Programmg 39 7-9. Orde A. (963). Mmzato of defte uadratc fuctos wth lear costrats. Recet advaces mathematcal programmg McGraw Hll. Rtter K. (966).A method for solvg maxmum problems wth a o-cocave uadratc obectve fucto. Z. Wahrschelchketstheore Verw. Geb. 4 340-35. Werers B.(987). A Iteractve fuzz programmg sstem. Fuzz sets ad sstems 33-47. Werers B. (987). Iteractve multple obectve programmg subect to flexble costrats. Europea Joural of Operatoal Research 3 34-349. YagT. J.P. Igzo ad Hu-Joo Km (99). Fuzz programmg wth o lear membershp fuctos: pecewse lear approxmato. Fuzz sets ad sstem 4 39-53. 4
Ye Yu (99). O affe scalg algorthms for o-covex uadratc programmg. Mathematcal Programmg 56 85-300. ZadehL.A. (965). Fuzz sets. Iformato ad Cotrol 8 338-353. Zmmerma H.J. (978). Fuzz programmg ad lear programmg wth several obectve fuctos. Fuzz Sets ad Sstems 45-55. ShashAggarwal s curretl workg as a Assocate Professor (o deputato) Cluster Iovato Cetre Uverst of Delh Ida. She s the permaet facult of the Departmet of Mathematcs Mrada House Uverst of Delh Ida. She receved her doctorate degree the feld of Mathematcal Programmg from Uverst of Delh Ida 993. She has publshed her research papers leadg ourals lke Europea Joural of Operatoal Research Optmzato Asa Pacfc Joural of Optmzato Opsearch Ida Joural of Pure ad Appled mathematcs Joural of Iformato & Optmzato Scece etc. Uda Sharma receved her B.Sc ad M.Sc from Uverst of Delh Ida 008 ad 00 respectvel. He s curretl dog Ph.D uder the supervso of frst author. 5