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1 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 QUADRATIC I-LEVEL PROGRAMMIG PROLEM ASED O FUZZY GOAL PROGRAMMIG APPROACH Surapat Pramank and Partha Pratm Dey Department of Mathematcs, andalal Ghosh.T. College, Panpur, P.O.-arayanpur, Dstrct - orth 4 Parganas, Pn Code-7436, est engal, Inda sura_pat@yahoo.co.n Patpukur Pallsree Vdyapth,, Pallsree Colony, Patpukur, Kolkata , est engal, Inda parsur.fuzz@gmal.com ASTRACT Ths paper presents fuzzy goal programmng approach to quadratc b-level programmng problem. In the model formulaton of the problem, we construct the quadratc membershp functons by determnng ndvdual best solutons of the quadratc objectve functons the system constrants. The quadratc membershp functons are then transformed nto equvalent lnear membershp functons by frst order Taylor seres approxmaton at the ndvdual best soluton pont. Snce the objectves of upper and lower level decson makers are potentally conflctng n nature, a possble relaxaton of each level decsons are consdered by provdng preference bounds on the decson varables for avodng decson deadlock. Then fuzzy goal programmng approach s used for achevng hghest degree of each of the membershp goals by mnmzng devatonal varables. umercal examples are provded n order to demonstrate the effcency of the proposed approach. KEYORDS -level programmng, Fuzzy programmng, Fuzzy goal programmng, Mult-objectve Quadratc programmng, Quadratc b-level programmng. ITRODUCTIO In ths paper, we consder quadratc b-level programmng problem (QLPP. QLPP conssts of a sngle decson maker namely upper level (frst level decson maker (ULDM wth sngle objectve at the upper level and a sngle decson maker namely lower level (second level decson maker (LLDM wth sngle objectve at the lower level. The objectve functon of each level decson maker (DM s quadratc n nature and the constrants are lnear functons. The executon of decson s sequental from upper level to lower level. Each level DM ndependently controls only a set of decson varables. The decson of ULDM s affected by the reacton of the LLDM due to dssatsfacton wth the decson of the ULDM. Therefore, decson deadlock arses frequently n the herarchcal organzaton n the decson-makng stuaton. -level programmng s a powerful and robust technque for solvng herarchcal decsonmakng problem. It has been appled n many real lfe problems such as agrculture, bo-fuel producton, economc systems, fnance, engneerng, bankng, management scences, transportaton problem, etc. The b-level programmng problem (LPP has receved ncreasng attenton n the lterature. Candler and Townsley [] as well as Fortuny- Amat and McCarl [] presented the formal formulaton of LPP. Anandalngam [3] proposed Stackelberg soluton concept to mult-level programmng problem (MLPP as well as b-level decentralzed programmng problem (LDPP. La [4] appled the concept of fuzzy set theory at frst to DOI : 0.5/jsea

2 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 MLPP by usng tolerance membershp functons. Shh et al. [5], Shh and Lee [6] extended La s concept by ntroducng non-compensatory max-mn aggregaton operator and compensatory fuzzy operator respectvely for MLPP. Sakawa et al. [7] developed nteractve fuzzy programmng for MLPP. Snha [8, 9] presented alternatve mult-level programmng based on fuzzy mathematcal programmng. Arora and Gupta [0] presented nteractve fuzzy goal programmng approach for lnear LPP wth the characterstcs of dynamc programmng. Satsfactory soluton s derved by updatng the satsfactory degree of the decson makers wth the consderaton of overall satsfactory balance between both the levels. A bblography of references on b-level programmng as well as mult-level programmng n both lnear and nonlnear cases, whch s updated bannually, can be found n the work of Vcente and Calama []. on-lnear LPP has been addressed n [-3]. Edmund and ard [3] dscussed nonlnear blevel mathematcal problems n 99. In contrast to LPPs, nonlnear LPPs [4, 5] have not been dscussed extensvely. Malhotra and Arora [6] developed an algorthm for solvng lnear fractonal b-level programmng problem (LFLPP based on preemptve goal programmng. Sakawa & shzak [7-8] used nteractve fuzzy programmng for solvng LFLPP both n crsp and fuzzy envronment. Calvete and Galé [9] studed optmalty condtons for LFLPP. Ahlatcoglu and Tryak [0] developed two nteractve fuzzy programmng algorthms for decentralzed two-level lnear fractonal programmng problem by usng the technque of multobjectve lnear fractonal programmng problem due to Chakraborty and Gupta [], and Charnes and Cooper []. Mshra [3] dscussed weghtng method for LFLPP by usng analytcal herarchy process [4]. However, the soluton obtaned by Mshra s approach [3] s the soluton of the ndvdual best soluton of the ULDM or the LLDM. It s worth mentonng that quadratc problems arse drectly n applcatons related to leastsquare regresson wth bounds or lnear constrants, central economc plannng, robust data fttng, nput-allocaton problem, transportaton, faclty locatons, traffc assgnment problem, portfolo optmzaton, etc. QLPP has been studed n [5-30] Vcente et al. [7] ntroduced two descent methods for QLPP n whch the lower level functon s strctly convex quadratc, the upper level functon s quadratc, and they proved that checkng local optmalty for blevel programmng s a P-hard problem. ang et al. [8] presented optmalty condtons and algorthm solvng lnear quadratc programmng problem. Thrwan and Arora [9] developed an algorthm for solvng QLPP for nteger varables. They solved the problem by lnearzaton technque and obtaned nteger soluton of the QLPP by usng Gomory cut and dual smplex method. Calvete and Galé [30] studed optmalty condtons for the lnear fractonal/ quadratc b-level programmng problem based on Karush Kuhn Tucker condtons and dualty theory. arang and Arora [3] presented an algorthm for solvng an ndefnte nteger QLPP wth bounded varables. They solved the problem by solvng the relaxed problem and developed a mxed nteger cut soluton technque for fndng the nteger soluton. Etoa [3] presented a smoothng sequental quadratc programmng to determne a soluton of a convex QLPP. L and ang [33] dscussed lnear-qlpp n whch the objectves of lower level are convex quadratc functons and the objectves of upper level are lnear functons. They transformed the orgnal problem nto equvalent non-lnear problem based on Karush Kuhn Tucker condtons and solved the equvalent problem usng genetc algorthm. Mshra and Ghosh [34] studed nteractve fuzzy programmng approach to b-level quadratc fractonal programmng problems by updatng the satsfactory level of the DM at the frst level wth consderaton of overall satsfactory balance between the levels. In fuzzy envronment, Pal and Motra [35] proposed fuzzy goal programmng (FGP procedure for solvng QLPP n 003. In [35], Pal and Motra formulated QLPP n two phases by usng the noton of dstance functon. At the frst phase of the soluton process, Pal and Motra transform QLPP model nto nonlnear goal programmng model n order to maxmze the membershp value of each of the 4

3 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 fuzzy objectve goals based on ther prortes n the decson context. However, each level DM has only one objectve functon, therefore the concept of prorty s not approprate. Recently, Pramank and Dey [36] studed prorty based FGP approach to mult-objectve quadratc programmng problem. In ths study, we extend the concept of Pramank and Dey [36] for solvng QLPP. e frst construct quadratc membershp functon by determnng ndvdual best soluton of the objectve functon of the level DMs. The quadratc membershp functons are then transformed nto lnear membershp functons by frst order Taylor seres approxmaton. Snce the objectves of the level DMs are generally conflctng n nature, possble relaxatons of decson of upper and lower level DMs are smultaneously consdered for avodng decson deadlock n the decson-makng stuaton by provdng preference bounds on the decson varables under ther control. Then FGP models are formulated for achevng hghest degree of each of the membershp goals by mnmzng negatve devatonal varables. To demonstrate the effcency of the proposed FGP approach, three numercal examples are solved and dstance functon s used to select compromse optmal soluton. Our man results are as follows: ( Two FGP models for solvng QLPP are presented. ( Maxmzaton-type and mnmzaton-type QLPPs are solved to demonstrate the applcablty of the proposed FGP models. ( Logcal explanatons are provded for consderng preference bounds on the decson varables. (v e transform the quadratc membershp functons nto equvalent lnear membershp functons at the ndvdual best soluton pont by frst order Taylor seres before usng FGP. Rest of the paper s organzed n the followng way: secton presents related works. Secton 3 provdes the formulaton of QLPP for maxmzaton-type objectve functon. In secton 4, we descrbe fuzzy programmng formulaton of QLPP. Subsecton 4. explans the lnearzaton of membershp functons by frst order Taylor polynomal seres. Subsecton 4., explans why DMs offer preference bounds on decson varables. Subsecton 4.3 presents formulaton of FGP models to QLPP. Secton 5 s devoted to provde formulaton of QLPP for mnmzatontype objectve functon. Secton 6 presents dstance functons to select compromse optmal soluton for the level DMs. Secton7 provdes FGP algorthm to QLPP. In secton 8, we solve three numercal examples n order to show the effcency of the proposed FGP approach. Fnally, secton 9 concludes the paper wth fnal concluson and future work.. RELATED ORKS FGP approach studed by Mohamed [37] s an mportant technque n dealng wth conflctng objectves of decson makers for satsfyng decson for overall beneft of the organzaton. Motra and Pal [38] extended the concept of Mohamed for solvng lnear LPP. Pramank and Roy [39] dscussed FGP approach to MLPP and they extended the FGP approach for a LDPP. They perform senstvty analyss wth the varaton of tolerance values on decson varables to show how the soluton s senstve to the change of tolerance values. aky [40] extended the concept of Motra & Pal [38] and Pramank & Roy [39] for solvng mult-objectve mult-level programmng problem. For non-lnear LPP, as already mentoned, QLPP was studed by Pal and Motra [35]. In ther approach, they formulate fuzzy quadratc programmng model to mnmze the group regret of degree of satsfacton of level decson makers by usng Hammng dstance [4]. Then they transform the quadratc model nto an equvalent non-lnear FGP model to acheve the hghest degree of satsfacton to the extent possble for the level decson makers. In the decson makng process, lnear approxmaton technque sutable for non-lnear goal programmng studed by Ignzo [4] s appled to obtan satsfactory soluton. Fnally, they formulate prorty based FGP model takng decson varables at frst prorty level and objectve goals at second prorty level wthout consderng the system constrants. They argued for not ncorporatng 43

4 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 system constrants n the fnal formulaton by statng that they do not come as a part of the decson search system. However, n actual practce t s observed that system constrants play vtal role for decson search systems. Therefore, due to neglectng system constrants ther FGP creates the problem of offerng nfeasble soluton or undesrable solutons. Agan, ther procedure ncludes several stages and transformaton varables as well as negatve and postve devaton varables that ndcate extra burden for solvng QLPP. Osman et al.[43] extended the fuzzy approach of Abo-Snna [5] for solvng non-lnear b-level and tr-level mult-objectve decson makng under fuzzness. Ther method based on the concept that the lower level decson maker maxmzes membershp goals takng a goal or preference of the ULDM nto consderaton. The level DMs elct non-lnear membershp functons of fuzzy goals for ther non-lnear objectve functons and especally the ULDM specfes lnear fuzzy goals for the decson varables. LLDM solves a fuzzy programmng wth a constrant on a satsfactory degree of ULDM. However, there s a possblty that ther fuzzy approach offers undesrable soluton because of nconsstency among the fuzzy goals of the non-lnear objectve functons and lnear fuzzy goals of the decson varables [44]. The research presented n ths paper ams to present easy and smple FGP algorthms for solvng QLPP by reducng complexty of transformaton varables and membershp goals of decson varables. 3. FORMULATIO OF QLPP e consder QLPP of maxmzaton - type of objectve functon at each level. Let ULDM controls the decson vector (x, x,...,x and LLDM controls the decson vector x (x, x,...,x. x Mathematcally, the problem can be formulated as: T [ULDM]: max Z ( C x x Dx ( x T [LLDM]: max Z ( C x x D x ( x x S {, x Ax A x, x 0, x 0 } (3 The symbol T denotes transposton. x x x s the set of decson vector, total number of decson varables and M s the total number of constrants n the system. C, C and are constant vectors. A, A are constant matrces. D, D are constant symmetrc matrces. e assume that the objectve functons are concave. Here, we also assume that the polyhedron S to be non-empty and bounded. 4. FUZZY PROGRAMMIG FORMULATIO OF QLPP To formulate the fuzzy programmng model of a QLPP, we transform the objectve functons Z (x and Z (x nto fuzzy goals by means of assgnng aspraton level to each of them. The optmal soluton of each objectve functon Z (x (,, when calculated n solaton, would be consdered as the best soluton and assocated objectve value can be consdered as the aspraton level of the correspondng fuzzy goal for -th level DM. x,x,...,x, x,..., x Let, (, be the ndvdual best soluton of the objectve functon of -th level DM the system constrants. 44

5 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 Also let, Z Z max Z x S (,. Then the fuzzy goals appear n the form: Z Z, Z Z ~ ~ Here ndcates the fuzzness of the aspraton level. ~ Usng the ndvdual best solutons, we formulate the payoff matrx as gven below: Z (x Z(x x Z(x Z(x (4 x Z(x Z(x The maxmum value of each column of payoff matrx provdes upper tolerance lmt or aspred level of achevement for the objectve functon.e. Z Z x max Z (, and the x S mnmum value of each column provdes lower tolerance lmt or lowest acceptable level of achevement for -th objectve functon.e. Z mnmum of {Z, Z } (,. The membershp functon of the ULDM can be wrtten as:, f Z(x Z Z(x Z, f Z Z(x Z Z Z 0, f Z(x Z Here, Z and goal for the ULDM. Smlarly, the membershp functon of the LLDM can be wrtten as: Z are respectvely the upper and lower tolerance lmts of the fuzzy objectve (5, f Z(x Z Z(x Z,f Z Z(x Z (6 Z Z 0, f Z(x Z Here, Z and Z are respectvely the upper and lower tolerance lmts of the fuzzy objectve goal for the LLDM. ow, the QLPP reduces to the followng problem: max (x (7 max (x (8 x S {, x Ax A x, x 0, x 0 }. 4.. Lnearzaton of membershp functons by Taylor seres approxmaton Let, x (x, x,...,x, x,...,x be the ndvdual best soluton of (x of -th level DM (, the system constrants. Then, we transform the quadratc membershp functon (x (, nto an equvalent lnear membershp functon ξ (, at 45

6 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 x (x,x,...,x,x,...,x by usng frst order Taylor seres approxmaton as follows: x - x x - x x - x x x x - x x... - x x ξ, (9 x - x x - x x - x x - x x - x x x ξ (0 4.. Characterzaton of preference bounds on the decson varables for both level DMs Snce the ndvdual best soluton of each level DM s dfferent, the queston of drect compromse optmal soluton does not arse. Therefore, cooperaton between the level DMs s necessary to reach a compromse optmal soluton. In ths context, each level DM tres to obtan maxmum beneft by consderng the beneft of other DM also. Therefore, we consder the relaxaton on decson of both the level DMs smultaneously to reach a compromse optmal soluton. In the proposed FGP approach, DMs provde ther preference upper and lower bounds x (j,,, be on the decson varables under ther control. Let and ( the lower and upper bounds of decson varable j a j j a j x j (j,,, provded by the ULDM. Here, x (x, x,...,x, x,...,x s the ndvdual best soluton of the membershp functon (x of UDLM when calculated n solaton the gven constrants. Smlarly, and ( j a j x (j,,, be the lower and upper bounds of decson varable j a j x j (j,,, provded by the LLDM. (x,x,...,x, x x,..., x s the ndvdual best soluton of the membershp functon (x of LLDM when calculated n solaton the gven constrants.therefore, we have ( j a j ( x j x j a j j Here, a j and x (j,,, ( x ( j a j x ( x (j,,, ( j a j a j (j,,, are the negatve and postve tolerance values, whch are not necessarly same. Generally, x j les between and ( j a j x (j,,,. j a j Smlarly, preference bounds of the decson varables under the control of LLDM can be determned Formulaton of FGP model of QLPP The QLPP reduces to the followng problem: max ξ (x (3 max ξ (x (4 46

7 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 S {(x,x Ax A x, x 0, x 0}, x ( j a j ( x j x j a j j x, (j,,, x ( j a j x ( x (j,,,. j a j The maxmum value of a membershp functon s unty, so for the defned membershp functons n (3 and (4, the flexble membershp goals wth aspraton level unty can be stated as: ξ x d d (5 ξ ( d d d Here, d represent the negatve devatonal varables and d, d represent the postve devatonal varables. In ths paper, we have consdered two FGP models for solvng QLPP. FGP model (: mn α d d (7 x - x x - x x - x - x x... - x x x d - d, x - x x - x x - x - x x - x x S {(x, x Ax A x, x 0, x 0}, ( j a j ( x j x j a j j d 0, d 0, x, (j,,, x ( j a j x ( x, (j,,, j a j d d 0, (,. x d - d, x x x FGP model (: mn γ (8 x - x x - x x - x - x x... - x x d - d, x - x x - x x - x - x x - x x x S {(x,x Ax A x, x 0, x 0}, ( x j a j j x, (j,,, x ( j a j x d - d, x x x x (6 47

8 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 x j a j x j j a j, (j,,, ( γ d, γ d, (, d 0, d 0, d d 0, (,. Snce the maxmum possble value of membershp goal s unty, postve devaton s not possble. Observng ths fact, Pramank and Dey used only negatve devatonal varable n the achevement functon [45, 46] e.g. (5 can be wrtten ξ x d (9 ( However, they do not mpose any restrcton on negatve devatonal varable. If we see (9, we observe that maxmum value of d wll be unty. Therefore, we have (0 Then accordng to Pramank and Dey [45, 46], and usng the restrcton (0, the proposed FGP models for solvng QLPP can be presented as: FGP model (I: mn α d ( x - x x - x x - x - x x... - x x x d, x - x x - x x - x - x x - x x S {(x,x Ax A x, x 0, x 0}, ( j a j ( x j x j a j j (,. x, (j,,, x ( j a j x ( x, (j,,, j a j x d, x x x FGP model (II: mn γ ( x - x x - x x - x x x - x x... - x x x d, x - x x - x x - x - x x - x x S {(x,x Ax A x, x 0, x 0}, ( j a j ( x j x j a j j γ d, (,, x, (j,,,, x ( j a j x ( x, (j,,,, j a j x d, x x 48

9 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 (,. 5. FORMULATIO OF QLPP FOR MIIMIZATIO-TYPE OJECTIVE FUCTIO Here, we consder QLPP for mnmzaton - type of objectve functon at each level. Mathematcally, QLPP can be presented as follows: T [ULDM]: mn Z ( C x x Dx x (3 T [LLDM]: mn Z ( C x x D x x (4 x S {, x Ax A x,x 0,x 0 } (5 Here, we assume that the objectve functons are convex and the polyhedron S s non-empty and bounded. Let, x (x, x,...,x, x,...,x (, be the ndvdual best soluton of -th level DM the gven constrants such that Z mn goals assume the form as Z (x Z (,. ~ x S Z (x (,. Then the fuzzy Usng the ndvdual best (mnmum solutons, we construct a payoff matrx as: Z (x Z(x x Z(x Z(x x Z(x Z(x The mnmum value of each column of Z (x (, gves lower tolerance lmt for the objectve functon.e. Z Z x mn Z (, and the maxmum value of each x S column provdes upper tolerance lmt for -th objectve functon.e. Z maxmum of {Z, Z } (,. The quadratc membershp functon for mnmzaton type objectve functon Z (x (, s formulated as: 0, f Z (x Z Z Z (x ν,f Z Z (x Z, (, (7 Z Z, f Z (x Z Here, Z and Z (, are the lower and upper tolerance lmts of the fuzzy objectve goal for -th level DM. The theoretcal concept of mnmzaton -type QLPP remans the same as dscussed for the maxmzaton - type QLPP. The proposed FGP models for solvng QLPP can be presented as: FGP model (I: mn α d (8 (6 49

10 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 ν x - x ν x - x ν x - x ν x - x ν x... - x x ν x d, ν x - x ν x - x - x ν x - x x ν x - x ν x d, ν x x S {, x Ax A x, x 0, x 0 }, ( j a j ( x j x j a j j x, (j,,, x ( j a j x ( x, (j,,, j a j (,. FGP model (II: mn γ (9 ν x - x ν x - x ν x - x ν x - x ν x... - x x ν x d, ν x - x ν x - x - x ν x - x x S {, x Ax A x, x 0, x 0 }, j a j xj ( j a j ( x ( x j a j j γ d, (, (,. x, (j,,, x, (j,,, j a j ν x - x ν x d, x ν x 6. USE OF DISTACE FUCTIOS TO OTAI COMPROMISE OPTIMAL SOLUTIO In the context of seekng optmal compromse soluton, t may be mentoned here that, n general, dfferent models (methods or approaches provde dfferent optmal solutons. Snce the objectve goals are conflctng n nature, the decson makers feel confused to select the best compromse soluton derved from dfferent models. In order to overcome such dffcultes, the concept of dstance functon ntroduced by Yu [47] can be used for measurng the deal pont dependent soluton for dentfyng the most satsfcng soluton. The famly of dstance functons for obtanng compromse optmal soluton s formulated as: 50

11 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 p L p ( ( d /p (30 Here, d (, denotes the degree of closeness of the preferred compromse soluton to the optmal soluton vector wth respect to -th objectve functon and the symbol p denotes the dstance parameter. For p, L ( ( d /. (3 ow for mnmzaton problem, d s defned as: d (ndvdual best soluton/ preferred compromse soluton and for maxmzaton problem, d s defned as: d (preferred compromse soluton/ ndvdual best soluton. The soluton for whch L ( ( d / s mnmum would be the compromse optmal soluton for the DMs. 7. THE FGP ALGORITHM FOR QLPP y the followng steps, we now present the proposed FGP algorthm for solvng QLPP: Step : Calculate the ndvdual best soluton x, x,...,x, x,..., x (, of each objectve functon Z (, for both the DMs the gven constrants. Step : Construct the payoff matrx. Then determne upper tolerance lmt and lower tolerance lmt of each objectve functon Z (, for -th level DM. Step 3: Construct the quadratc membershp functon (x or ν (x (, of the fuzzy objectve goal for each level DM. Step 4: Determne the ndvdual best soluton x (x,x,...,x,x,...,x of the quadratc membershp functon (x or ν (x (, of -th level DM (, the constrants. Step 5: Transform the quadratc membershp functon (x (, nto equvalent lnear membershp functon ξ (, at the ndvdual best soluton pont x (x,x,...,x,x,...,x by usng frst order Taylor seres approxmaton as gven by (9 and (0. Step 6: oth level DMs provde ther preference upper and lower bounds on the decson varables. Step 7: Formulate the FGP models (FGP model (I and FGP model (II. Step 8: Solve the models. Step 9: Dstance functon L s used to dentfy the compromse optmal soluton for both level DMs. Step 0: End. 8. UMERICAL EXAMPLES Example. To llustrate the proposed FGP approach, we consder the followng problem wth maxmzaton type of objectve functon at each level: [ULDM]: max Z 6x 3x - x - x [LLDM]: x x x 5, 3x x 9, max Z x 5x - x x 5

12 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 x x 6, x 0, x 0. e fnd the ndvdual best soluton Z at (.308,.038 and Z at (.556,.67 the constrants for ULDM and LLDM respectvely. Then the fuzzy goals appear n the form: Z and Z ~ ~ Payoff matrx Here, Z 8.79 and Z The quadratc membershp functons of both level DMs are constructed as: Z (x (6x 3x - x - x -8.79, Z (x (x 5x - x The membershp functon (x for ULDM s maxmal at the pont (.308,.038 and the membershp functon (x for LLDM s maxmal at the pont (.556,.67. Then the quadratc membershp functons are transformed nto equvalent lnear membershp functons at the ndvdual best (maxmal soluton pont by frst order Taylor polynomal seres as follows: ξ (x (.308,.038 (x (.308,.038 (x (.308,.038, x x ξ (x (.556,.67 (x (.556,.67 (x -.67 (.556,.67. x x Let.5 x 3 and.5 x 3 be the preference bounds provded by the respectve level DMs. Then the proposed FGP models can be wrtten as: FGP model (I: mn α d (x (0.73 (x (0.488 d, (x (.795 (x -.67 (.96 d, x x 5, 3x x 9, x x 6,.5 x 3, 0.5 x 3, (,, x 0, x 0. Then, followng the procedure, the proposed FGP model (I gves the soluton at x.75, x The membershp values are ξ 0.999, ξ. FGP model (II: mn γ (x (0.73 (x (0.488 d, (x (.795 (x -.67 (.96 d, x x 5, Z 9.96, Z 5

13 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 3x x 9, x x 6,.5 x 3, 0.5 x 3, γ d, (,, (,. x 0, x 0. The proposed FGP model (II provdes the soluton Z 0.397, The correspondng membershp values are ξ 0.999, ξ Z at x.53, Table. Comparson of dstances for the optmal solutons of the numercal Example based on proposed two FGP models. Proposed models x, x Z, Z L FGP model (I.75, , FGP model (II.53, , x ote : From table, we observe that the proposed FGP model (II offers better optmal soluton than the proposed FGP model (I based on dstance functon L by consderng same preference bounds. Example. e consder the followng QLPP studed by Pal and Motra [35]: [ULDM]: max Z x x - (x - x [LLDM]: max Z (x - x x x x 6, x x, -x x, -x x, x, x 0. e fnd the ndvdual best soluton Z 36 at (4, and Z 6 at (, 4 the constrants for ULDM and LLDM respectvely. Then the fuzzy goals appear n the form: Z 36 and Z 6. ~ ~ 36 8 Payoff matrx 6 6 Here, Z 6 and Z 8. The quadratc membershp functons of both level DMs are constructed as: Z (x - 6 x x (x, Z (x -8 (x x The membershp functon (x for ULDM s maxmal at the pont (4, and the membershp functon (x for LLDM s maxmal at the pont (, 4. Then the quadratc membershp 53

14 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 functons are transformed nto equvalent lnear membershp functons at the ndvdual best (maxmal soluton pont by frst order Taylor polynomal seres as follows: ξ (x (4, (x - 4 (4, (x - (4,, x x ξ (x (, 4 (x - (,,4 (x - 4 (4,. x x Let 3 x 5 and x 6 be the preference bounds provded by the respectve level DMs. Then the proposed FGP models can be wrtten as: FGP model (I: mn α d (x - 4 7/30 d, (x - 4 d, x x 6, x x, -x x, -x x, 3 x 5, x 6, (,, x 0, x 0. Then, followng the procedure, the proposed FGP model (I gves the soluton at x 3, x 3. The membershp values are FGP model (II: mn γ (x - 4 7/30 d, (x - 4 d, x x 6, x x, -x x, -x x, 3 x 5, x 6, γ d, (,, (,. x 0, x ,.5 The proposed FGP model (II provdes the same soluton set Z 30, Z 0 at Z 30, x 3, Z 0 x 3. The membershp values are.467,.5. Pal and Motra [35] obtaned the same soluton set. Usng the same tolerance for x and for x, as consdered n the proposed FGP models, Osman et al. [43] obtaned leader s ndvdual best soluton (4, whch cannot be acceptable for the lower level decson maker. Example3. e consder the followng QLPP wth mnmzaton type of objectve functon at each level: 54

15 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 [ULDM]: mn Z 3 x 4 x - x - x x [LLDM]: mn Z 5 x x - x - x x x x, x 3x 7, x 0, x 0. The ndvdual best (mnmum soluton for ULDM and LLDM are Z 0.58 at (0.789, 0.4 and Z 0.75 at (0.5, respectvely the constrants. The fuzzy goals are as follows: Z 0.58 and Z ~ ~ The payoff matrx s of the form Here, Z and Z.836. The quadratc membershp functons for ULDM and LLDM are of the form:.75 Z(x.75 (3x 4x x x ν and Z(x.836 (5x x x x ν The quadratc membershp functon for ULDM s maxmal at (0.789, 0.4 the constrants and the quadratc membershp functon for LLDM s maxmal at (0.5, the constrants. The quadratc membershp functons ν and ν are transformed nto equvalent lnear membershp functons at the ndvdual maxmal pont as follows: ξ (x ν (0.789, 0.4 (x ν(0.789,0.4 (x -0.4 ν(0.789, 0.4, x x ξ (x ν (0.5, (x -0.5 ν (0.5, (x - ν (0.5, x x Let 0.55 x.5 and 0.5 x. be the preference bounds provded by ULDM and LLDM respectvely. The proposed FGP models for solvng QLPP can be formulated as: FGP model (I: mn α d (x (-.77 (x 0.4 ( d, (x 0.5 ( (x (-.84 d, x x, x 3x 7, 0.55 x.5, 0.5 x., (,, x 0, x 0. Solvng the above FGP model (I, the soluton set s obtaned as 0.75, x 0.5. The resultng membershp values are ξ 0.999, FGP model (II: mn γ Z 0.88, ξ. Z.56 at x 55

16 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 (x (-.77 (x 0.4 ( d, (x 0.5 ( (x (-.84 d, x x, x 3x 7, 0.55 x.5, 0.5 x., γ d, (,, (,. x 0, x 0. y solvng the FGP model (II we get the same soluton set x 0.5. The obtaned membershp values are ξ 0.999, ξ. Z 0.88, Z.56 at x 0.75, ote : From Example 3, we see that the proposed FGP model (I and FGP model (II offer the same soluton set the same preference bounds. ote 3: e observe that the proposed two FGP models offer the same soluton set or dfferent soluton set dependng on the problem consdered. Therefore, t s better to solve the problems by both the FGP models and use dstance functon L to dentfy the compromse optmal soluton. ote 4: All solutons of the problem are obtaned by Lngo software verson COCLUSIOS In ths paper, an alternatve FGP approach has been studed for solvng QLPP. The proposed approach s easy to mplement. Frstly, we transform QLPP nto a lnear b-level programmng problem by usng frst order Taylor seres approxmaton. Preference bounds provded by the upper and lower level DMs are consdered for relaxaton on decson. Then two FGP models are formulated n order to solve the problem by mnmzng negatve devatonal varables. Here, we do not requre postve devatonal varables. e can apply the proposed concept to mult-level quadratc and mult-level quadratc fractonal programmng problem. The proposed concept can also be extended to solve QLPP wth fuzzy parameters. The man drawback of the proposed approach s that t solves hypothetcal problem. Here, for decson-makng, degrees of membershp functons of the objectve goals are consdered. However, ut, degree of rejecton should be smultaneously consdered. In ths sense, ntutonstc fuzzy sets due to Atanassov [48] and ntutonstc fuzzy goal programmng technque due to Pramank and Roy [49-5] could be appled to modelng QLPP after usng lnearzaton technque. e hope that the proposed FGP approach can contrbute to future study n the feld of practcal herarchcal decson-makng problems nvolvng quadratc objectves especally n ndustral, marketng, supply-chan management problems, etc. Our future work wll nclude the use of the concept presented n ths paper to develop an algorthm for solvng lnear fractonal / quadratc b-level programmng problem. Fnally, t s worth mentonng that, although the proposed FGP approach s frutful and easy to mplement n dealng wth QLPP, t s not the only approach to be taken to solve QLPP. Specal attenton has also to be pad n dealng wth QLPP n ntutonstc fuzzy envronment. Research n the feld nvolvng QLPP n ntutonstc fuzzy envronment s, therefore, an open ssue. 56

17 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 ACKOLEDGEMETS The authors are very grateful to the anonymous referee for her helpful comments and constructve suggestons for mprovng the qualty and presentaton of the paper to ts current standard. The authors are very grateful to Dr.. C. Gr for hs constant encouragements and constant support durng the present study. REFERECES [] Candler,. & Townsley, R., (98 A lnear blevel programmng problem, Computers & Operatons Research, Vol. 9, o., pp [] Fortun-Amat, J. & McCarl,., (98 A representaton and economc nterpretaton of a two-level programmng problem, Journal of Operatonal Research Socety, Vol. 3, o. 9, pp [3] Anandalngam, G., (988 A mathematcal programmng model of decentralzed mult-level systems, Journal of the Operatonal Research Socety, Vol. 39, o., pp [4] La, Y. J., (996 Herarchcal Optmzaton: A satsfactory soluton, Fuzzy Sets and Systems, Vol. 77, o. 3, pp [5] Shh, H. S., La, Y. J. & Lee, E. S., (996 Fuzzy approach for mult-level programmng problems, Computers & Operatons Research, Vol. 3, o., pp [6] Shh, H. S. & Lee, E. S., (000 Compensatory fuzzy multple level decson makng Fuzzy Sets and Systems, Vol. 4, o., pp [7] Sakawa, M., shzak, I. & Uemura, Y., (998 Interactve fuzzy programmng for mult-level lnear problems, Computers and Mathematcs wth Applcatons, Vol. 36, o., pp [8] Snha, S., (003 Fuzzy programmng approach to mult-level programmng problems, Fuzzy Sets and Systems, Vol. 36, o., pp [9] Snha, S., (003 Fuzzy mathematcal programmng appled to mult-level programmng problems, Computers Operatons Research, Vol. 30, o. 9, pp [0] Arora, S. R., & Gupta, R., (009 Interactve fuzzy goal programmng approach for blevel programmng problem, European Journal of Operatonal Research, Vol. 94, o., pp [] Vcente, L., & Calama, P. H., (994 level and multlevel programmng: a bblography revew Journal of Global Optmzaton, Vol. 5, o. 3, [] Chen, Y., Floran, M., (995 The non-lnear blevel programmng problem: formulatons, regularty and optmalty condtons Optmzaton, Vol. 3, o. 3, [3] Edmunds, T. & ard, J., (99 Algorthms for nonlnear blevel mathematcal problems, IEEE Transactons Systems, Man, and Cybernetcs, Vol., o., pp [4] Savard, G. & Gauvn, J., (994 The steepest descent drecton for the nonlnear blevel programmng problem, Operatons Research Letters, Vol. 5, o. 5, pp [5] Abo-Snna, M. A., (00 A b-level non-lnear mult-objectve decson makng under fuzzness Journal of the Operatonal Research Socety of Inda (OPSEARCH, Vol. 38, o. 5, pp [6] Malhotra,. & Arora, S. R., (000 An algorthm to solve lnear fractonal blevel programmng problem va goal programmng, Journal of Operatonal Socety of Inda (OPSEARCH, Vol. 37, o., pp [7] Sakawa, M. & shzak, I., (00 Interactve fuzzy programmng for two-level lnear fractonal programmng problem, Fuzzy Sets and Systems, Vol. 9, o., pp [8] Sakawa, M., shzak, I., & Uemura, Y., (000 Interactve fuzzy programmng for two-level lnear fractonal programmng problems wth fuzzy parameters Fuzzy Sets and Systems, Vol. 5, o., [9] Calvete, H. I. & Galé, C., (004 Solvng lnear fractonal blevel programs, Operatons Research Letters, Vol. 3, o., pp [0] Ahlatcoglu, M. & Tryak, F., (007 Interactve fuzzy programmng for decentralzed two-level lnear fractonal programmng (DTLLFP problems, Omega, Vol.35, o. 4, pp [] Chakraborty, M., & Gupta, S., (00 Fuzzy mathematcal programmng for mult objectve lnear fractonal programmng problem Fuzzy Sets and Systems, Vol. 5, o. 3, pp [] Charnes, A., & Cooper,.., (96 Programmng wth lnear fractonal functons aval Research Logstcs Quarterly, Vol. 9, o., [3] Mshra, S., (007 eghtng method for b-level lnear fractonal programmng problems, European Journal of Operatonal Research, Vol. 83, o., pp

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19 Internatonal Journal of Software Engneerng & Applcatons (IJSEA, Vol., o.4, October 0 [50] Pramank, S. & Roy, T. K., (007 Intutonstc fuzzy goal programmng and ts applcaton n solvng mult-objectve transportaton problem, Tamsu Oxford Journal of Management Scences, Vol., o., pp [5] Pramank, S. & Roy, T. K., (007 An ntutonstc fuzzy goal programmng approach for a qualty control problem: a case study, Tamsu Oxford Journal of Management Scences, Vol. 3, o. 3, pp Authors DR. SURAPATI PRAMAIK He obtaned Ph. D. n Mathematcs from engal Engneerng and Scence Unversty, Shbpur. He s the Assstant Professor of Mathematcs n andalal Ghosh.T. College. He s a senor lfe member of Operaton Research Socety of Inda, ISI (Kolkata and Calcutta Mathematcal Socety. He receved Golakpat Roy Memoral Award n 988. Hs nterest of research ncludes Fuzzy optmzaton, Grey system theory, Intutonstc fuzzy decson makng, eutrosophc sets, and Math educaton. Hs paper together Partha Pratm Dey was awarded best paper n est engal State Scence and Technology Congress (SSTC-0 n mathematcs. Hs paper together wth Manjra Saha was awarded best paper n SSTC-00 n Socal Scences. Hs paper together wth Sourendranath Chakrabart and T. K. Roy was awarded best paper n SSTC-008 n mathematcs. He publshed research papers n Internatonal Journal of Computer Applcatons, European Journal of Operatonal Research, Journal of Transportaton Systems Engneerng and Informaton Technology, Tamsu Oxford Journal of Management Scences, otes on Intutonstc Fuzzy Sets. MR. PARTHA PRATIM DEY Date of rth: ovember 3, 98. He passed. Sc Honours and M. Sc n Mathematcs n 003 and 005 respectvely n the Unversty of Kalyan. Currently, he s the assstant teacher of mathematcs at Patpukur Pallsree Vdyapth, Patpukur, est engal. Hs felds of research nterest nclude Fuzzy optmzaton, Decentralzed b-level decson makng, Grey system theory, IFS sets, and eutrosophc sets. Hs paper together wth S. Pramank was awarded best paper n est engal Scence and technology Congress -0 n Mathematcs. 59