Chapter 2 A Class of Robust Solution for Linear Bilevel Programming

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1 Chapter 2 A Class of Robust Soluton for Lnear Blevel Programmng Bo Lu, Bo L and Yan L Abstract Under the way of the centralzed decson-makng, the lnear b-level programmng (BLP) whose coeffcents are supposed to be unknown but bounded n box dsturbance set s studed. Accordngly, a class of robust soluton for lnear BLP s defned, and the orgnal uncertan BLP was converted to the determnstc trple level programmng, then a solvng process s proposed for the robust soluton. Fnally, a numercal example s shown to demonstrate the effectveness and feasblty of the algorthm. Keywords Box dsturbance Lnear blevel programmng Robust optmzaton Robust soluton 2.1 Introducton Blevel programmng (BLP) s the model wth leader-follower herarchcal structure, whch makes the parameter optmzaton problems as the constrants (Dempe 2002). In ts decson framework, the upper level programmng s connected wth not only the decson varables n ts level but also wth the optmal soluton n the lower level programmng, whle the optmal soluton n the lower lever programmng s affected by decson varables n the upper level B. Lu B. L Y. L School of Management, Tanjn Unversty, Tanjn, Chna B. Lu (&) School of Informaton Scence and Technology, Shhez Unversty, Xnjang, Chna e-mal: lubo.ce@163.com Y. L School of Scence, Shhez Unversty, Xnjang, Chna E. Q et al. (eds.), The 19th Internatonal Conference on Industral Engneerng and Engneerng Management, DOI: / _2, Ó Sprnger-Verlag Berln Hedelberg

2 12 B. Lu et al. programmng. Due to the leader-follower herarchcal structure problems wdely exst n the realstc decson-makng envronment, the scholars have been payng great attenton to BLP and have brought about good results on the theory and algorthms (Balas and Karwan 1982; Fortuny-Amat and McCarl 1981; Matheu et al. 1994; La 1996). Some degree of uncertanty exsts n realstc decsonmakng envronment, such as the nevtable error of measurng nstrument n data collecton, ncompleteness n data nformaton, the approxmate handle for the model and other factors; hence t s necessary to study on the uncertan Blevel programmng. For the uncertanty problem, the fuzzy optmzaton and stochastc optmzaton have been appled wdely. However, t s dffcult for decson-makers to gve the precse dstrbuton functons or membershp functons whch are requred n above methods. Thus, the robust optmzaton become an mportant method, because t can seek for the best soluton for the uncertan nput wthout consderng the parameter dstrbuton of uncertan parameters and s mmune from the uncertan data (Soyster 1973). For the uncertan BLP, the defnton of robust soluton s nfluenced by the dependent degree of the upper and lower levels n the decson-makng process. When the dependent degree s relatve ndependence, the robust soluton to the uncertan BLP s defned by the way of the decentralzed decson-makng (L and Du 2011); when the dependent degree s relatve dependence, the robust soluton to the uncertan BLP s defned by the way of the centralzed decson-makng, that s, when the lower level seeks ts own robust soluton, t consders the nfluence to the robust soluton of the upper level frstly. In the paper, the latter case wll be dscussed, and the coeffcents of BLP are supposed to be unknown but bounded n box dsturbance set. By the transform of the uncertan model, the robust soluton of BLP s obtaned. Fnally, a numercal example s shown to demonstrate the effectveness and feasblty of the algorthm. 2.2 The Defnton of Robust BLP The Model and the Defnton In ths paper we consder Lnear BLP formulated as follows: mn Fðx; yþ ¼c T x 1 x þ dt 1 y s:t: where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Ax þ By h x; y 0 ð2:1þ

3 2 A Class of Robust Soluton for Lnear Blevel Programmng 13 In model (2.1), x 2 R m1 ; y 2 R n1 ; c l 2 R m1 ; d l 2 R n1 ; l 2f1; 2g; A 2 R rm ; B 2 R rn ; h 2 R r1 ; there s some uncertanty or varaton n the parameters c 1 ; d 1 ; c 2 ; d 2 ; A; B; h. Let ðc 1 ; d 1 ; c 2 ; d 2 ; A; B; hþ 2l, l s a gven uncertanty set n Box dsturbance as follows: 8 >< l :¼ ðc l ; d l ; A; B; hþ >: j 2 c l ¼ c l þ ð u c l d lj ¼ dlj þ ð u d l a k ¼ a k þ ð u A b kj ¼ b kj þ ð u B h k ¼ h k þ ð u h Þ ; ðu cl Þ j ; ðu dl Þ k ; ðu A Þ kj ; ðu B Þ ð u c l Þ j ð u d l Þ ðu cl Þ Þ j ðu dl Þ j ð u AÞ k ðu A ð u BÞ kj ðu B Þ k ; ðu h Þ k ð u hþ k ðu h l ¼f1; 2g; 2 f1;...; mg; f1;...; ng; k ¼ f1;...; rg: Þ k 9 >= >; ð2:2þ For l ¼f1; 2g; 2 f1;...; mg; j 2 f1;...; ng; k ¼ f1;...; rg, c l ; d lj ; a k ; d kj ; h k are the gven data, and ðu cl Þ ; ð u d l Þ j ; ð u A ; ð u B ; ð u hþ k are the gven nonnegatve data. Under the way of the centralzed decson-makng, the robust soluton of uncertan BLP (1) s defned as follows: Defnton 1 (1) Constrant regon of the lnear BLP (1): X ¼ fðx; yþjax þ By h; x; y 0; ða; B; hþ 2lg (2) Feasble set for the follower for each fxed x XðxÞ ¼fyAxþ j By h; x; y 0; ða; B; hþ 2lg (3) Follower s ratonal reacton set for each fxed x ( ( MðxÞ ¼ yy2arg mn ct 2 x þ )) dt 2 y; y 2 XðxÞ; ða; B; hþ 2l (4) Inducble regon: IR ¼ fðx; yþjðx; yþ 2X; y 2 MðxÞg:

4 14 B. Lu et al. Defnton 2 Let ( F :¼ ðx; y; tþ 2R m R n c T 1 R x þ dt 1 y t ) ðx; yþ; ðc 1 ; d 1 Þ2l The programmng mnftjðx; y; tþ 2Fg ð2:3þ x;y;t s defned as robust counterpart of uncertan lnear BLP(1); F s defned as the robust feasble set of uncertan lnear BLP(1) The Transform of Uncertan BLP Model Under the way of the centralzed decson-makng, based on the orgnal dea of robust optmzaton that the objectve functon can get the optmal soluton even n the worst and uncertan stuaton, the transform theorem can be descrbed as followngs: Theorem The robust lnear BLP (1) wth ts coeffcents unknown but bounded n box dsturbance set l s equvalent to Model (2.4) wth certan coeffcents as followngs: mn x Fðx; yþ ¼ Xm s:t: where d 2 solves X m max c 1 þ l c d 1 2 c 1 þ l c 1 d1j þ l d 1 d1j þ l d 1 j y j s:t: d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j ; j ¼f1;...; ng; where y solves mn f ðx; yþ ¼d T y 2 y s:t: Xm a k ð l A k ¼ f1;...; rg x; y 0: b kj ð l B j y j y j h k þðl hþ k ; ð2:4þ Proof (1) Frstly, the constrant regon X of the lnear BLP (1) s transformed nto the certan regon. Consder the constrant regon of the lnear BLP (1):

5 2 A Class of Robust Soluton for Lnear Blevel Programmng 15 X ¼ fðx; yþjax þ By h; x; y 0; ða; B; hþ 2lg Accordng to the process of the transformaton (Lobo et al. 1998), we can obtan Ax þ By h; ða; B; hþ 2l 8 ><, 0 mn l A ;l B ;l h, 0 Xm >< þ mn l A ;l B ;l h >: X m a k x þ Xn 8 >: X m a k x þ Xn b kj y j h k ðu A Þ k x þ Xn x;y 0, 0 Xm a k ð u A 8k2f1;...;rg, Xm a k ð u A a k ¼ a k þ u 9 ð AÞ k ; ðu A ð u AÞ k ðu A ; b kj ¼ b kj þ ð u BÞ kj ; ðu B b kj y j h ð u BÞ kj ðu B ; >= k h k ¼ h k þ ð u hþ k ; ðu h Þ k ð u hþ k ðu h Þ k ; 2f1;...; mg; j 2f1;...; ng; >; k 2f1;...; rg ðu B ðu A ð u AÞ k ðu A Þ 9 k ðu B ð u BÞ kj ðu B >= Þ k ðu h Þ k ð u hþ k ðu h Þ k 2f1;...; mg; j 2f1;...; ng; >; k 2f1;...; rg b kj ð u B y j h k þ ð u h Þ kj y j ðu h b kj ð u B Þ k y j h k þ ð u hþ k ; k 2f1;...; rg So the lnear BLP (1) s transformed nto the model (2.6) as followngs: ð2:5þ

6 16 B. Lu et al. mn Fðx; yþ ¼c T x 1 x þ dt 1 y s:t: where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Xm a k ð u A k ¼ f1;...; rg x; y 0 b kj ð u B y j h k þ ð u h Þ k ð2:6þ (2) Next, accordng the equvalent form (Lobo et al. 1998) mn f ðxþ x s:t: x 2 D, mn x;t t s:t: f ðxþt x 2 D and the K-T method, the model (2.6) can be transform-ed nto the model (2.7) (L and Du 2011): mn Fðx; yþ ¼ t x;t s:t: c T 1 x þ dt 1 y t where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Xm a k ðu AÞ k k ¼ f1;...; rg x; y 0 b kj ðu B y j h k þðu hþ k ð2:7þ (3) Smlar to the transformaton (2.5), c T 1 x þ dt 1 y t; ðc x;y 0 Pm 1; d 1 Þ2l, c 1 þ l c 1 x þ Pn d 1j þ l d 1 j So the model (2.7) can be transformed to the model (2.8) as follows: y j t

7 2 A Class of Robust Soluton for Lnear Blevel Programmng 17 mn Fðx; yþ ¼ Xm c x;t 1 þ l c 1 s:t: where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Xm a k ð u A k 2f1;...; rg; x; y 0 b kj ð u B d1j þ l d 1 j y j y j h k þ ð u hþ k ; ð2:8þ (4) Next, because the optmal soluton of BLP (1) s not nfluenced by the value of c 2, we only consder how to choose the value of d 2. Based on the orgnal dea of robust optmzaton, the model (2.8) s transformed nto the model (2.4) above. 2.3 Solvng Process of the Model The determnstc trple level programmng (2.4) can be wrtten as the followng programmng (2.9) by the K-T method. mn x max d 2 ;y;u;v s:t: d 2j ¼ Xr Fðx; yþ ¼Xm u k b kj ð u B k¼1 c 1 þ l c 1 þ v j d1j þ l d 1 j y j d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j " # X m u k a kj ð l AÞ x þ Xn b kj ð l B y j ðh k þ ð l hþ k Þ ¼ 0 ¼j v j y j ¼ 0 X m a k ð l A j ¼ f1;...; ng; k ¼ f1;...; rg; x; y; u; v 0: b kj ð l B y j h k þ ð l hþ k ; ð2:9þ Accordng to the lterature (Wang 2010), the model (2.9) can be transformed nto the model (2.10) as follows

8 18 B. Lu et al. mn t x;d 2 ;y;u;v s:t: Xm d 2j ¼ Xr c 1 þ l c 1 u k k¼1 b kj ð u B d1j þ l d 1 þ v j ; j y j t d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j " ; # X m u k a k ð l A b kj ð l B y j h k þ ð l h ¼ 0; v j y j ¼ 0; X m a k ð l A j ¼ f1;...; ng; k ¼ f1;...; rg; x; y; u; v 0: b kj ð l B y j h k þ ð l hþ k ; Þ k ð2:10þ By ntroducng a large constant M, the model (2.10) above can be transformed nto a mxed nteger programmng as follows (Fortuny-Amat and McCarl 1981): mn t x;d 2 ;y;u;v;t;w s:t: Xm c 1 þ l c 1 d 2j ¼ Xr u k k¼1 b kj ð u B d1j þ l d 1 þ v j ; d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j ; y j Mt j ; v j M 1 t j ; u k Mw k ; X m a k ð l A b kj ð l B X m a k ð l A h b kj ð l B j y j t y j h k þ ð l h Þ k y j h k þ ð l hþ k ; j ¼ f1;...; ng; k ¼ f1;...; rg; t j 2 f0; 1g; w k 2 f0; 1g; x; y; u; v 0: The model (2.11) can be solved by the software Lngo 9.0 M ð 1 wk Þ ð2:11þ

9 2 A Class of Robust Soluton for Lnear Blevel Programmng A Numercal Example We gve a numercal example to demonstrate the proposed approach as follows: mn F ¼ c 11 x þ d 11 y 1 þ d 12 y 2 x s:t: 2:5 x 8 where y 1 ; y 2 solve mn f ¼ d 21 y 1 þ d 22 y 2 y 1 ;y 2 s:t: a 11 x 1 þ b 11 y 1 þ b 12 y 2 h 1 a 21 x 1 þ b 21 y 1 þ b 22 y 2 h 2 a 31 x 1 þ b 31 y 1 þ b 32 y 2 h 3 a 41 x 1 þ b 41 y 1 þ b 42 y 2 h 4 y 1 0; y 2 0 where a 11 ¼ 0; b 21 ¼ 1; a 31 ¼ 0; a 41 ¼ 0; b 41 ¼ 0; b 42 ¼ 1: And the others are the uncertan data, the gven varables and dsturbances are c 11 ¼ 1:5; d 11 ¼ 1:5; d 12 ¼ 2; d 21 ¼ 1:5; d 22 ¼ 3:5; b 11 ¼ 1:75; b 12 ¼ 1:15; h 1 ¼ 2:5; a 21 ¼ 3:5; b 22 ¼ 1:5; h 4 ¼ 5:75: h 2 ¼ 11; b 31 ¼ 3:5; b 32 ¼ 1:25; h 3 ¼ 23; ðu c1 Þ 1 ¼ 0:5; ð u d 1 Þ 1 ¼ 0:5; ð u d 1 Þ 2 ¼ 3; ð u d 2 Þ 1 ¼ 0:5; ð u d 2 Þ 2 ¼ 0:5; ðu b ðu b Þ 11 ¼ 0:25; ð u bþ 12 ¼ 0:15; ð u hþ 1 ¼ 0:5; ðu a Þ 31 ¼ 0:5; ð u b Þ 21 ¼ 0:5; ð u b Þ 22 ¼ 0:5; ð u hþ 2 ¼ 1; Þ 32 ¼ 0:25; ð u hþ 3 ¼ 1; ð u hþ 4 ¼ 0:25: Accordng to the theorem and these data above, robust model transformed s demonstrated as

10 20 B. Lu et al. mn t x;y;u;v;g;z s:t: 2x y 1 þ y 2 t 2:5 x 8; y 2 5:5; 1:5y 1 1:3y 2 3; 4x þ y 1 2y 2 10; 4y 1 1:5y 2 22; 1 1:5u 1 þ u 2 4u 3 þ v 1 2; 4 1:3u 1 2u 2 1:5u 3 u 4 þ v 2 3; y 1 Mg 1 ; y 2 Mg 2 ; v 1 Mð1 g 1 Þ; v 2 Mð1 g 2 Þ; u k Mz k ; k ¼ 1; 2; 3; 4: 1:5y 1 1:3y 2 3 ð1 z 1 Þ; 4x þ y 1 2y 2 þ 10 ð1 z 2 Þ; 4y 1 1:5y 2 þ 22 ð1 z 3 Þ; y 2 þ 5:5 ð1 z 4 Þ; x; y; u; v; g; z 0; g 1 ; g 2 2 f0:1g; z k 2 f0:1g; k ¼ 1; 2; 3; 4: By the software Lngo 9.0, the robust soluton s obtaned as follows: ðx; y 1 ; y 2 Þ¼ð2:5201; 4:6443; 2:2819Þ; The robust optmal value s F mn ¼ 2: Concluson and Future Work Under the way of the centralzed decson-makng, a class of robust soluton for uncertan lnear BLP s defned, whch expands further the applcaton of BLP n dfferent crcumstances. And based on the orgnal dea of robust optmzaton, the uncertan BLP was converted to the determnstc trple level programmng. The solvng process s proposed to obtan the robust soluton of uncertan lnear BLP. Fnally, a numercal example s shown to demonstrate the effectveness and feasblty of the algorthm.

11 2 A Class of Robust Soluton for Lnear Blevel Programmng 21 References Balas WF, Karwan MH (1982) On two-level optmzaton. IEEE Trans Autom control AC- 27(1): Dempe S (2002) Foundatons of blevel programmng. Kluwer Academc Publsher, Boston Fortuny-Amat J, McCarl B (1981). A representaton and economc nterpretaton of two-level programmng problem. J Oper Res Soc 32: Fortuny-Amat J, McCarl BA (1981) Representaton and economc nterpretaton of two-level programmng problem. J Oper Res Soc 32(7): La YJ (1996) Herarchcal optmzaton: a satsfactory soluton. Fuzzy Sets and Syst 77: L Y, Du G (2011) Robust lnear blevel programmng under ellpsodal uncertanty. Syst Eng 11: Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applcaton of second-order cone programmng. Lnear Algebra Appl 284: Matheu R, Pttard L, Anandalngam G (1994) Genetc algorthm based approach to b-level lnear programmng. Oper Res 28:1 21 Soyster AL (1973) Convex programmng wth set-nclusve constrants and applcatons to nexact lnear programmng. Oper Res 21: Wang J (2010) Research on the methods of nterval lnear b-level programmng, pp Tanjn Unversty, Tanjn

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