A PROCEDURE FOR SOLVING INTEGER BILEVEL LINEAR PROGRAMMING PROBLEMS

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1 ISSN: Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology A ISO 397: 7 Certfed Orgazato) Vol. 3, Issue, Jauary 4 A PROCEDURE FOR SOLVING INTEGER BILEVEL LINEAR PROGRAMMING PROBLEMS A.Y. Adham, ad Quazzaf Rabba Assstat Professor, Departmet of Mathematcs, Itegral Uversty, Lucow, Ida, ABSTRACT: Ths paper s a eteso of the K th-best approach [4] for solvg blevel lear programmg problems wth teger varables. NAZ cut [] ad A-T cut [3] are added to reach the teger optmum. A eample s gve to show the effcecy of the proposed algorthm. Keywords: A-T cut, blevel lear teger programmg, K th-best approach, NAZ cut. I. INTRODUCTION Blevel programmg has bee proposed for dealg wth decso processes volvg two decso maers wth a herarchcal structure. A blevel programmg problem BLPP) cossts of two levels, amely, the frst level ad the secod level. The frst level decso maer s called the leader ad the secod s called the follower. The follower eecutes ts polces after ad vew of, the decsos of hgher level decso maer.e. leader. I terms of applcatos, blevel programmg has bee used may domas, e.g. to desg optmzato problems process system egeerg [6], desg of trasportato etwor [], agrcultural plag [9], maagemet of mult-dvsoal frms [4]. May researchers have desged algorthms for the soluto of the BLPP [, 4, 5, 8, ]. However, there has bee very lttle atteto the lterature o both the soluto ad the applcato of blevel problems volvg dscrete decsos. Ths s maly because these problems pose major algorthmc challeges the developmet of effectve soluto strateges. For the soluto of the purely teger lear BLPP, a brach ad boud type of eumeratve soluto algorthm has bee developed by Moore ad Bard []. Cuttg plae ad parametrc soluto approaches have bee developed by Dempe [7]. Sahards ad Ierapetrtou [5] gave a algorthm for the resoluto of med teger BLPP based o Beders decomposto method. I ths paper we focus o the teger lear blevel programmg problem, whch all volved fuctos are lear. The am of ths paper s to preset a eteded K th-best approach for fdg the teger soluto to a blevel programmg problem by troducg A-T cut to the reduced feasble rego obtaed after usg NAZ cut. II. DESCRIPTION OF NAZ CUT AND A-T CUT FOR INTEGER LINEAR PROGRAMMING PROBLEMS Cosder the pure lear teger programmg problem as follows: ma f, s. t. A, A ) c b c, are tegers The lear programmg relaato ca be obtaed by omttg the teger restrctos. Copyrght to IJIRSET )

2 ISSN: Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology A ISO 397: 7 Certfed Orgazato) Vol. 3, Issue, Jauary 4 Frst we solve the lear programmg relaato. Let the soluto be, ). If s all teger, the the problem s solved. th Let the compoet of be o teger wth a. The earest teger values to where ] are [ a ] ad [ ] { a a}, for,. [t s the largest teger less tha or equal to t ad {t} s the smallest teger greater tha or equal to t. Wth such bfurcatos we ca fd all the pots the surroudg of the o-teger soluto. Deote the set of dces of these pots by S. If all these pots le outsde the feasble rego we move to the et teger feasble pots obtaed from a. Let the objectve value at Now we fd the dfferece be Z. Thus, the objectve fucto level plae at d Z c, S, Copyrght to IJIRSET wll be c Z.,.e., the dfferece betwee the objectve fucto value at o teger soluto ad the objectve fucto values at the surroudg teger pots, as suggested by Rabba ad Adham [3]. Where ' s, S, are surroudg teger pots aroud. Now we search for the feasble pot, whch has a mmum postve dfferece from the objectve fucto value. Let G be the set of dces S for whch ' s are feasble. Let { d m d } G A plae passg through ths teger pot ad parallel to the objectve hyperplae wll be c Z. Z Z Clearly The NAZ cut s ow troduced as c Z whch reduces the feasble rego. Here Z acts as a lower boud for the teger soluto to the problem. Let be defed as:,,, ); Now to fd the teger optmum soluto we add the A-T cut at j j S j j. S as III. THE PROCEDURE Usg the commo otato blevel programmg, the teger lear blevel programmg ILBP problem ca be wrtte as follows: ma f, ) c c, where solves,

3 ISSN: Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology ma f, ) A ISO 397: 7 Certfed Orgazato) Vol. 3, Issue, Jauary 4 d ) s.t. A A b,,,, are tegers where c s a -dmesoal row vector, c ad d are - dmesoal row vectors, A s a m -matr ad b s a m -dmesoal colum vector. We assume that the polyhedro S defed by the commo costrats s oempty ad bouded. Frstly we solve the lear programmg LP relaato for leader s problem assocated wth ) usg smple method.e., we solve, ma f, c c Let the soluto be passes through problem at. ) s.t. A A b, a),. If the soluto s o teger we add the NAZ cut c c c c z,, s the teger pot sde the feasble rego ad, where ) j j Now to fd the teger optmum soluto we add the A-T cut Let that [ ], [], [ N ] S j j at. whch z s the value of leader s deote the N ordered basc feasble solutos to the ILBP for a) such c,,, N ). c[ ] Let S be the projecto of S oto the leader s decso space. For each [ ] S, a feasble soluto to the ILBP problem ) s obtaed by solvg the followg teger lear programmg problem: ma d s.t. A b A, b) ad teger. [] For the above problem also we ca fd the teger optmum by usg NAZ cut ad A-T cut. Let M ) deote the set of optmal soluto to b). We assume that for ay fed choce of leader, follower has some room to respod,.e., M ). Hece, the feasble rego of the leader, called the ducble rego IR, s [ ] IR=, ) : S, M )}. { [] Wth the above etesos the K th-best algorthm we ca fd the teger optmum soluto for the blevel programmg problems. [] The procedure ca be summarzed the followg steps: Copyrght to IJIRSET 835

4 ISSN: Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology A ISO 397: 7 Certfed Orgazato) Vol. 3, Issue, Jauary 4 Step. Set. Solve a) wth the smple method. If the soluto s o teger the add NAZ cut ad A-T cut to obta teger optmum soluto as []. Let W ) ad T. Go to Step. Step. Solve b) for teger optmum soluto usg NAZ cut ad A-T cut. Let ths soluto be deoted by ~. If ~, stop; s the global optmum to ). Otherwse, go to Step 3. [ ] Step 3. Let W deote the set of adjacet etreme pots of Let T T ) ad W W W ) T ] Step 4. Set ad choose Cosder the followg ILBP problem: ma f c [. Go to step 4. such that Copyrght to IJIRSET so that c ma W c)., ) 8 Go to step. IV. NUMERICAL EXAMPLE c. c [ ] where solves: ma f, ) 3) s.t ,,, are tegers. The frst step of the above procedure s to solve the lear programmg problem ma f, ) s.t. 4 3a), We get the o teger soluto as. 66,. 36 ad f, ) We roud off the o teger soluto to the earest four teger pots as, ),, 3), 3, ) ad 3, 3). The respectve dffereces are ; ; ; We are left wth oly oe feasble pot, ), whch gves the mmum postve dfferece. Now the NAZ cut ad A-T cut passg through the teger pot, ) ca be derved respectvely as ad Now solvg the problem 3) wth these addtoal costrats we obta the teger optmum soluto as, 4 ad f, ) 88,4, the frst best soluto. Set W {,4)} ad T. Let ) [ ]

5 ISSN: Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology A ISO 397: 7 Certfed Orgazato) Vol. 3, Issue, Jauary 4 To determe f s a elemet of M ) [] [] we solve ma f, ) s.t b) 84,, teger. After addg the requred NAZ cut ad A-T cut we get the teger optmal soluto as ~,4). Hece, ~ [ ] Therefore,,4) s the global optmal soluto to ILBP problem 3). V CONCLUSION We have eteded the Kth-best algorthm for solvg lear blevel programmg problems wth the help of NAZ cut for teger programmg alog wth the A-T cut. Ths algorthm gves us the teger soluto for blevel programmg problems wth much computatoal ease. REFERENCES [] J.F. Bard, A Algorthm for Solvg the Geeral Blevel Programmg Probem. Mathematcs of Operatos Research. Vol. 8, No., pp. 6-7, 983. [] A. Bar, ad Q. S. Ahmad, NAZ cut for Iteger Programmg. Pure ad Appled Mathemata Sceces. Vol LVII, No.-, 87-94, 3. [3] A. Bar, ad T. Alam, Search for Iteger Optmum after Addg NAZ cut. Pure ad Appled Mathemata Sceces. Vol. LXII, No. -, pp , 5. [4] W. F. Balas, ad M. H. Karwa, Two Level Lear Programmg, Maagemet Scece, Vol. 3, No. 8, pp.4-, 984. [5] M Campelo ad S. Schemberg, A Smple Approach for Fdg Local Solutos of a Lear Blevel Program by Equlbrum Pots Aals of Operatos Research. Vol. 38, pp , 5. [6] P. A. Clar ad A.W.Westerberg, Blevel Programmg for Steady-State Chemcal Process Desg I. Fudametals ad Algorthms. Comput. Chem. Eg. Vol. 4, No., pp , 99. [7] S. Dempe, DscreteBlevelOptmzatoProblems.http;// dempe, TU Chemtz. [8] N.P. Faísca, V. Dua, B. Rustem, P.M. Sarava, E.N. Pstopoulos, Parametrc Global Optmzato for Blevel Programmg. Joural of Global Optmzato. Vol. 38, pp , 7. [9] J. Fortuy-Amat, B. McCarl, A Represetato ad Ecoomc Iterpretato of a Two-Level Programmg Problem. Joural of Operatos Research Socety. Vol. 3, No. 9, pp , 98. [] J. Judce, ad A. Fausto, A Sequetal LCP method for Blevel Lear Programmg. Aals of Operatos Research. Vol. 34, pp. 89-6, 99. [] L.J. LeBlac, ad D.E. Boyce, A Blevel Programmg Algorthm for Eact Soluto of Networ Desg Problem wth User-Optmal Flows. Tras. Res.-Part B, Vol., pp.59-65, 985. [] J.T. Moore, ad J.F. Bard. The Med Iteger Lear Blevel Programmg Problem. Operatos Research. Vol. 38, pp. 9-9, 99. [3] Q. Rabba, ad A.Y. Adham, A Note o NAZ cut for Iteger Progrmmg Pure ad Appled Mathmata Sceces, Vol. LXVIII, No. -, pp , 8 [4] J. Ryu, V. Dua, E.N. Pstopoulos, A Blevel Programmg Framewor for Eterprse-Wde Process Networs uder ucertaty. Comput. Chem. Eg. Vol. 8, pp. -9, 4. [5] G.K. Sahards, ad M. G. Ierapetrtou, Resoluto Method for Med Iteger Blevel Lear Problems Based o Decomposto Techque. J. Glob. Optm. 8. Copyrght to IJIRSET 837