International Journal of Pure and Applied Sciences and Technology

Transcription

1 Int. J. Pure Appl. Sc. Technol., 6( (0, pp. 5-3 Internatonal Journal of Pure and Appled Scences and Technology ISS Avalable onlne at Research Paper Goal Programmng Approach to Lnear Fractonal Blevel Programmng Problem Based on Taylor Seres Approxmaton Partha Pratm Dey and Surapat Pramank, Patpukur Pallsree Vdyapth,, Pallsree Colony, Patpukur, Kolkata , West Bengal, Inda. Department of Mathematcs, andalal Ghosh B.T. College, Panpur, P.O.- arayanpur, Dstrct orth 4 Parganas, Pn Code-7436, West Bengal, Inda. Correspondng author, e-mal: (sura_pat@yahoo.co.n (Receved: -7-; Accepted: 9-9- Abstract: Ths paper deals wth lnear fractonal blevel programmng problem. The goals of objectve functons are determned by optmzng ndvdual objectve functon the system constrants. Then the fractonal objectve functons are transformed nto equvalent lnear functons at the optmal soluton pont by usng frst order Taylor polynomal seres. Snce the objectves of the decson makers are potentally conflctng n nature, decson makers consder relaxaton on decson for avodng decson deadlock. To demonstrate the effcency of the proposed approach, a numercal example s solved and compared wth other approaches. Keywords: Goal programmng, Fractonal programmng, Blevel programmng, Lnear fractonal blevel programmng, Taylor seres.. Introducton: Blevel programmng [6, 7,, 5] comprses of the objectve functon of the upper (frst level decson maker (UDM at ts frst level and that of the lower (second level decson maker (LDM at the second level. The executon of decson power s sequental from upper level to lower level. In the decson makng stuaton, each decson maker (DM ndependently controls only a set of decson varables. Candler and Townsley [] as well as Fortuny-Amat and McCarl [5] were credted to formulate the formal blevel programmng problem (BLPP. Anandalngam [4] studed multlevel programmng problem (MLPP based on Stackelberg soluton concept and extended the concept for decentralzed blevel programmng problem (DBLPP. A bblography of reference on

2 Int. J. Pure Appl. Sc. Technol., 6( (0, BLPP as well as MLPP n both lnear and nonlnear cases, whch s updated bannually, can be found n the work of Vcente and Calama [9]. Fractonal programmng (FP s a specal case of nonlnear programmng. Guzel and Svr [6] presented Taylor seres soluton for mult-objectve lnear fractonal programmng problem (MOLFPP n crsp envronment. Toksarı [8] proposed Taylor seres approach to fuzzy MOLFPP. Thrwan and Arora [6] studed lnear fractonal blevel programmng problem (LFBLPP n 993. Thrwan and Arora [5] proposed an algorthm for the nteger LFBLPP n 997. In 995, Mathur and Pur [8] stated that the optmal soluton of the LFBLPP mght occur at a non-extreme pont. In 998, Calvete and Galé [] showed that the example posed by Mathur and Pur [8] s not well posed because the second level problem has multple optma. In 999, Calvete and Galé [0] developed an enumeratve algorthm, whch offers a global optmal soluton to the blevel lnear/lnear fractonal programmng problem. Arora et al. [5] presented an algorthm for LFBLPP when the follower controls few varables. Malhotra and Arora [7] used goal programmng (GP approach for solvng LFBLPP. Alemayehu and Arora [3] dscussed nteger LFBLPP. Calvete and Galé [8, 9] dscussed optmalty condtons for LFBLPP. Sakawa and shzak [, 3] used nteractve fuzzy programmng (IFP for solvng LFBLPP. Adoptng the same concept of Sakawa and shzak [], Mshra and Ghosh [0] dscussed IFP approach to blevel quadratc fractonal programmng problem. Ahlatcoglu and Tryak [] developed two dfferent IFP algorthms for decentralzed LFBLPP based on the technque of MOLFPP due to Chakraborty and Gupta [3] and Charnes and Cooper [4]. Mshra [9] dscussed weghtng method for LFBLPP by usng analytcal herarchy process [4]. However, the optmal soluton obtaned by Mshra [9] s the ndvdual best soluton of the UDM. Abo-Snna and Baky [] dscussed fuzzy goal programmng procedure to blevel MOLFPP by usng the concept of MOLFPP due to Pal et al. []. Toksarı [7] developed Taylor seres approach for solvng LFBLPP due to Guzel and Svr [6] and Toksarı [8]. Toksarı [7] obtaned the optmal soluton whch s the ndvdual best soluton of the LDM. In general, the ndvdual best soluton of LDM cannot be acceptable to the UDM. To overcome such problems, we use GP approach to LFBLPP by consderng that DMs provde ther preference bounds on the decson varables under ther control. To compare the effcency of our proposed approach dstance functon [30] s used. The performance of the proposed GP approach s expermentally valdated by an example. Rest of the paper s structured as follows. Secton descrbes the constructon of LFBLPP. Secton 3 provdes GP formulaton of LFBLPP and soluton procedure. Secton 4 presents performance analyss. Secton 5 provdes numercal example to llustrate the soluton procedure. Secton 6 presents analyss of obtaned results. Secton 7 presents the concluson.. Constructon of the LFBLPP: UDM: max X LDM: max ( Χ Χ = ( a Χ a Χ c / ( b Χ b Χ d ( = ( a Χ a Χ c / ( b Χ b Χ d ( ( X, X S = {( X,X AX A X B, X 0,X 0} (3

3 Int. J. Pure Appl. Sc. Technol., 6( (0, Here, a, b ( =, are dmensonal constant row vectors and a, b ( =, are -dmensonal constant row vectors and c, d ( =, are constants. A ( =, s an M ( =, constant matrx. Here, we assume that mn{( b Χ b Χ d Χ S}> 0 ( =, (4 S s assumed to be non-empty, convex and compact set n R. Let X = ( X,X R and =. Here, decson vector Χ ( =, s an -dmensonal decson varables of the -th level DM and X = (x, x,, x. For the sake of smplcty, we consder the followng addtonal notatons a = ( a, a, b = ( b, b and A = [ A, A ], then a compact form of LFBLPP can be wrtten as UDM: max Χ = ( a Χ c / ( b Χ d (5 LDM: max Χ ( ( Χ Χ = ( a Χ c / ( b Χ d (6 Χ S = { Χ R ΑΧ Β, Χ 0 } (7 Here, a, b ( =, are -dmensonal constant row vectors, c, d ( =, are constants and Β s an M-dmensonal constant column vector; Α s an M constant matrx; and mn{( b d Χ S}> 0 ( =,. (8 Χ Here, S s assumed to be non-empty, convex, and compact n R. 3. GP formulaton of LFBLPP by Taylor seres approxmaton: x = ( x,x,...,x,x,..., x = Let (x ;, where ( =, be the best ndvdual soluton of the objectve functon of -th level DM where max Χ ( =, (9 = ( =,. In the proposed GP approach, the level DMs provde ther preference upper and lower bounds x (j =,,, be Then, the objectve goal assumes the form ( Χ X S on the decson varables under ther control. Let, ( x and ( j r j ( j r j the lower and upper bounds of decson varables provded by the UDM where x s the ndvdual best soluton of the objectve functon of UDM the system constrants S. x (j =,,, be the lower and upper bounds of Smlarly, ( x and ( j r j j r j decson varables provded by the LDM where x s the ndvdual best soluton of the objectve functon of LDM the system constrants S.

4 Int. J. Pure Appl. Sc. Technol., 6( (0, Therefore, we have ( x x j ( x (j =,,, and ( x x j ( j r j. Here, rj and j r j j r j x (j =,,, j r j r j ( =,, (j =,,, are the negatve and postve tolerance varables. Generally, x j les between ( x r and ( r j j x ( =,, (j =,,,. j j We transform the fractonal objectve functon nto equvalent lnear objectve functon at the ndvdual best soluton pont by frst order Taylor seres approxmaton. Then we apply GP to solve the problem. By the followng steps, we now explan the proposed GP approach. x = ( x,x,...,x,x,..., x Step: Fnd whch s the ndvdual best soluton of the objectve functon of -th level DM ( =,, where s the number of varables. Step: The goal of objectve functon of -th level DM, ( =, s determned X S ( Χ by = max ( =,. Step3: Transform fractonal objectve functon nto equvalent lnear objectve functon by usng frst order Taylor seres approxmaton as follows ( Χ x ( x - x (x - x ( X ( Χ x x = ( Χ x ( x - x (x - x ( X x (x - x (x - x x at X = x... (x - x, (0 x x = ( Χ x (x - x (x - x x at X = x (x - x. ( Step4: Let the bounds provded by the respectve level DMs be x (j =,,,, ( ( x j r j x j ( j r j ( x x j ( j r j x (j =,,,. (3 j r j Step5: Form the GP Model as Mn α (4

5 Int. J. Pure Appl. Sc. Technol., 6( (0, x ( x - x x x x ( x - x x (x - x (x - x x x... (x - x (x - x d - d =, (5 - x x x (x - x - ( x j r j x j ( j r j ( x x j ( j r j (x - x x x (x - x (x - x d - d =, (6 x (j =,,,, (7 j r j x (j =,,,, (8 α d ( =,, (9 ī α d ( =,, (0 - - d. d = 0 ( =,, d 0, d 0 ( =,, ( Χ S. ( Here - - d, d represent negatve devatonal varables and d, d represent postve devatonal varables. Step6: Solve the problem (4. If the soluton s acceptable to UDM and LDM, then optmal soluton s reached. Otherwse, UDM and LDM provde another set of preference bounds on the decson varables to obtan an optmal compromse soluton.e. go to step 5 untl the compromse soluton s reached. Step7: End. 4. Performance analyss: To compare the soluton wth other methods, the followng famly of dstance functons [30] s defned: p t t / p L p ( λ, t = T p λ ( d (3 t = Here, d t (t =,,, T represents the degrees of closeness of the preferred compromse soluton to the optmal soluton vector wth respect to the t-th objectve functon. Here, λ = ( λ,..., λt represents vector of attrbute attenton levels λ t. We assume that λt =. If T t =

6 Int. J. Pure Appl. Sc. Technol., 6( (0, all the attrbutes are equal, then λ = /T (t =,,, T. The power p represents the dstance t parameter p. / For p =, L ( λ, t = T t ( dt λ (4 t = For maxmzaton problem, d t s denoted by d t = (the preferred compromse soluton/ (the ndvdual best soluton. For mnmzaton problem, d t s denoted by d t = (the ndvdual best soluton/ (the preferred compromse soluton. The soluton for whch L ( λ, t wll be mnmal would be the most satsfyng soluton for UDM and LDM. Therefore, by comparng the dstance L ( λ, t, one can compare the performance of the solutons obtaned by dfferent approaches. 5. Example: We solve the followng numercal example taken from [9] wth changed constrants to demonstrate the soluton procedure and clarfy the effectveness of the proposed GP approach. UDM: max ( x =(x x /(x x (5 LDM: x max ( x x -x x 3, x -3x 3, x x 3, x, x 0. = (x x /(x 3x (6 = 0.7 at (.4, 0.6. We fnd the ndvdual best soluton =.3 at (5, 9 and Then the fractonal objectve functons are transformed nto equvalent lnear objectve functons by usng frst order Taylor polynomal seres. ( x (x -5 ( x ( 5,9 ( x at x = (5, x 0.07x = ( x (x -9 ( x at x = (5,9 = ( x (7 (8 ( x (x -.4 ( x (.4,0.6 at x = (.4,0.6 (x -0.6 ( x at x = (.4,0.6 = ( x (9

7 Int. J. Pure Appl. Sc. Technol., 6( (0, 5-3. ( x x x = ( x (30 Then the objectve goals assume the forms ( x =.3 and ( x = 0.7. Let the preference bounds provded by the respectve DMs be x 6, (3 8 x 6, (3 Then GP model can be wrtten as Mn α ( x 0.07x - d - d =.3, x x d - d = 0.7, x 6, 8 x 6, -x x 3, x -3x 3, x x 3, α d -, α d, α d -, α d, - d. d = 0, - d. d = 0, - x, x 0, d 0, d 0 ( =,. We obtan the optmal soluton of the problem (33 as x = 3.5, x = 8, =.3, = 0.673, L = ote. Followng [7], the optmal compromse soluton set obtaned as x =.4, x = 0.6, = 0.9, = 0.7, L = Here, t s to be noted that the optmal soluton obtaned from the method proposed by Toksarı [7], s actually the LDM s ndvdual best soluton that cannot be generally acceptable to UDM. ote. If the preference bounds provded by the respectve DMs be the same as we consder.e. x 6 and 8 x 6, Mshra s model [9] offers the ndvdual best soluton of UDM. Table o. : Comparson of optmal soluton obtaned from dfferent approaches Approach Bounds on Optmal L varables soluton x, x Proposed GP x 6, 3.5, Model 8 x 6 Toksarı [7] -.4, Mshra [9] x 6, 8 x 6 5,

8 Int. J. Pure Appl. Sc. Technol., 6( (0, Analyss of obtaned results: The weghtng method of Mshra [9] offers the soluton set as x = 5, x = 9 wth =.3, = 0.674, L = , whch s the ndvdual best soluton of UDM. UDM gets 00% of hs/her aspraton level. LDM gets 94.70% of hs/her aspraton level. Ths ndcates that the LDM has to accept the ndvdual best soluton of UDM. Ths soluton cannot be satsfactory for LDM as the UDM s alone fully satsfed to acheve hs /her aspraton level. Taylor seres approach of Toksarı [7] provdes the soluton set as x =.4, x = 0.6, = 0.9, = 0.7, L = 0.59, whch s the ndvdual soluton of LDM. UDM acheves 68.8% of hs/her aspraton level. LDM acheves 00% of hs /her aspraton level. Ths mples that UDM has to accept the ndvdual soluton of LDM. Ths leads to the paradox that the decson power of LDM domnates the decson power of UDM. In our proposed approach, GP Model offers the compromse soluton set as x = 3.5, x = 8, =.3, = 0.673, L = Here we observe that UDM gets 99.3% of hs/her aspraton level and LDM gets 94.80% of hs/her aspraton level. Herarchy s mantaned n achevng the aspred level of the goals and compromse soluton s obtaned as achevement of UDM s less than 00% and achevement of lower level DM s 94.80%. Comparng the dstance functon L (See Table o. we observe that we obtan better optmal soluton than Mshra [9] and Toksarı [7]. 7. Concluson: We present GP approach to LFBLPP n a smple way. We convert LFBLPP to an equvalent lnear BLPP by usng frst order Taylor polynomal seres. We do not need any extra transformaton varables. By comparng dstance functon L, we observe that our proposed GP approach offers better optmal soluton than proposed by Toksarı [7] and Mshra [9]. The proposed concept can also be extended to lnear fractonal multlevel programmng problem. We hope that the proposed approach can contrbute to future study n the feld of agrcultural system, producton plannng problems, etc. nvolvng fractonal objectves. Acknowledgements The authors want to thanks, Hemen Datta, Managng Edtor of IJPAST and anonymous referees for ther valuable suggestons. References [] M. A. Abo-Snna and I. A. Baky, Fuzzy goal programmng procedure to blevel multobjectve lnear fracton programmng problems, Internatonal Journal of Mathematcs and Mathematcal Scences, (00, 0-5, ID ( do:0.55/00/ [] M. Ahlatcoglu and F. Tryak, Interactve fuzzy programmng for decentralzed two-level lnear programmng (DTLLFP problems, Omega, 35 (4 (007, [3] G. Alemayehu and S. R. Arora, On the blevel nteger lnear fractonal programmng problem, Journal of the Operatonal Research Socety of Inda (OPSEARCH, 38 (5 (00, [4] G. Anandalngam, A mathematcal programmng model of decentralzed mult-level systems, Journal of the Operatonal Research Socety, 39 ( (988,

9 Int. J. Pure Appl. Sc. Technol., 6( (0, [5] S. R. Arora,. Malhotra and D. Thrwan, An algorthm for blevel fractonal program when the follower controls few varables, Indan Journal of Pure and Appled Mathematcs, 33 ( (00, [6] J. F. Bard, Optmalty condtons for the b-level programmng problem, aval Research Logstcs Quarterly, 3 ( (984, 3-6. [7] W. F. Balas and M. H. Karwan, Two level lnear programmng, Management Scence, 30 (8 (984, [8] H. I. Calvete and C. Galé, Optmalty condtons for the lnear fractonal/quadratc blevel problem, Monografas del Semnaro Matematco Garca de Galdeano, 3 (004, [9] H. I. Calvete and C. Galé, Solvng lnear fractonal blevel programs, Operatons Research Letters, 3 ( (004, [0] H. I. Calvete and C. Galé, The b-level lnear/lnear fractonal programmng problem, European Journal of Operatonal Research, 4 ( (999, [] H. I. Calvete and C. Galé, On the quasconcave blevel programmng problem, Journal of Optmzaton Theory and Applcatons, 98 (3 (998, [] W. Candler and R. Townsley, A lnear two-level programmng problem, Computers & Operatons Research, 9 ( (98, [3] M. Chakraborty and S. Gupta, Fuzzy mathematcal programmng for mult objectve lnear fractonal programmng problem, Fuzzy Sets and Systems, 5 (3 (00, [4] A. Charnes and W. W. Cooper, Programmng wth lnear fractonal functons, aval Research Logstcs Quarterly, 9 (3-4 (96, [5] J. Fortun- Amat and B. McCarl, A representaton and economc nterpretaton of a two-level programmng problem, Journal of the Operatonal Research Socety, 3 (9 (98, [6]. Guzel and M. Svr, Taylor seres soluton of multobjectve lnear fractonal programmng problem, Trakya Unversty Journal Scence, 6 ( (005, [7]. Malhotra and S. R. Arora, An algorthm to solve lnear fractonal blevel programmng problem va goal programmng, Journal of the Operatonal Research Socety of Inda (OPSEARCH, 37 ( (000, 0-3. [8] K. Mathur and M. C. Pur, On blevel fractonal programmng, Optmzaton, 35 (3 (995, 5-6. [9] S. Mshra, Weghtng method for b-level lnear fractonal programmng problems, European Journal of Operatonal Research, 83 ( (007, [0] S. Mshra and A. Ghosh, Interactve fuzzy programmng approach to b-level quadratc fractonal programmng problems, Annals of Operatonal Research, 43 ( (006, [] B. B. Pal, B.. Motra and U. Maulk, A goal programmng procedure for fuzzy multobjectve lnear fractonal programmng, Fuzzy Sets and Systems, 39 ( (003, [] M. Sakawa and I. shzak, Interactve fuzzy programmng for two-level lnear fractonal programmng problem, Fuzzy Sets and Systems, 9 ( (00, [3] M. Sakawa and I. shzak, Interactve fuzzy programmng for cooperatve two-level lnear fractonal programmng problems wth multple decson makers, Internatonal Journal of Fuzzy Systems, ( (999, [4] T. L. Satty, The AnalytcalHerarchy Pocess, Plenum Press, ew York, 980. [5] D. Thrwan and S. R. Arora, An algorthm for the nteger lnear fractonal blevel programmng problem, optmzaton,39 (( [6] D. Thrwan and S. R. Arora, B-level lnear fractonal programmng problem, Cahers Du Cero, 35 (- (993, [7] M. D. Toksarı, Taylor seres approach for b-level lnear fractonal programmng problem, Selçuk Journal of Appled Mathematcs, ( (00, [8] M. D. Toksarı, Taylor seres approach to fuzzy multobjectve lnear fractonal programmng, Informaton Scences, 78 (4 (008, [9] L.. Vcente and P. H. Calama, Blevel and multlevel programmng: a bblography revew, Journal of Global Optmzaton, 5 (3 (004, [30] M. eleny, Multple Crtera Decson Makng, McGraw Hll, ewyork, 98.