AN EXTENDED KTH-BEST APPROACH FOR REFERENTIAL-UNCOOPERATIVE BILEVEL MULTI-FOLLOWER DECISION MAKING

Transcription

1 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), AN EXTENDED TH-BEST APPROACH FOR REFERENTIAL-UNCOOPERATIVE BILEVEL MULTI-FOLLOWER DECISION MAING GUANGQUAN ZHANG, CHENGGEN SHI, JIE LU Facut of Inforaton Technoog Unvert of Technoog Sdne PO Box 23, Broadwa NSW 2007, Autraa Ea: {zhangg, ch, eu}@t.ut.edu.au Beve decon technque have been an deveoped for ovng decentrazed anageent probe wth decon aker n a herarchca organzaton. When utpe foower are nvoved n a beve decon probe, caed a beve ut-foower (BLMF) decon probe, the eader decon w be affected, not on b the reacton of thee foower, but ao b the reatonhp aong thee foower. The referenta-uncooperatve tuaton one of the popuar cae of BLMF decon probe where thee utpe foower don t hare decon varabe wth each other but a take other decon a reference to ther decon. Th paper preent a ode for the referenta-uncooperatve BLMF decon probe. A the kth-bet approach one of the ot uccefu approache n deang wth nora beve decon probe, th paper then propoe an extended kth-bet approach to ove the referenta-uncooperatve BLMF probe. Fna an exape of ogtc pannng utrate the appcaton of the propoed extended kth-bet approach. eword: Beve prograng, kth-bet approach, Decon akng, Optzaton.. Introducton In genera, a beve decon probe ha three portant feature: () there ext two decon unt wthn a predonant herarchca tructure; (2) the decon unt at the ower eve execute t poce after, and n vew of, a decon ade at the upper eve; (3) each unt ndependent optze t obectve but affected b the acton of other unt. The decon unt (decon aker) at the upper eve tered a the eader, and at the ower eve, the foower. The eader cannot copete contro the decon ade b h/her foower but nfuenced b the reacton of the foower. The opta outon of the foower aow the eader to copute h/her obectve functon vaue. Such a decon tuaton ha appeared n an decentrazed organzaton, and been an handed b near beve prograng (BLP) technque. A nuber of beve decon approache and agorth have been propoed to fnd an opta outon for a near beve decon probe, uch a the uhn- Tucker approach, 2 branch and bound approach, 3 the kthbet approach, 4 and other.,5,6 When a beve decon probe decrbed b a near BLP, at eat one opta (goba) outon can be attaned at an extree pont of the contrant regon. Th reut wa frt etabhed b Cander and Towne 7 wth no upper-eve contrant and wth unque ower eve outon. Afterward Bard 8 and Baa and arwan 9 proved th reut under the aupton of that the contrant regon bounded. The reut for the cae where the upper eve contrant ext wa etabhed b Savard 0 wthout an partcuar aupton. Baed on th reut, Cander and Towne 7 and Baa and arwan 9 propoed repectve the kth-bet approach that copute goba outon of near BLP probe b enueratng the extree pont of the contrant regon. The kth-bet Pubhed b Atant Pre Coprght: the author 205

2 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), G. Zhang, C. Sh, and J. Lu approach ha then been proven to be a vauabe ana too wth a wde range of uccefu appcaton for near BLP.,7,9 Our prevou work -3 extended the kthbet approach n handng a ore wde range of beve decon probe. In rea-word beve decon probe, the ower eve a nvove utpe ndependent decon unt, that, utpe foower. For exape, the CEO of a copan the eader and a drector of branche of th copan are the foower n akng a product deveopent pan. The eader (the CEO) decon w be affected, not on b the reacton of the utpe foower (thee drector of branche), but ao b the reatonhp aong thee foower. We ca uch a probe a beve ut-foower (BLMF) decon probe. Thee foower a do or a don t hare ther decon varabe, obectve or contrant. For exape, thoe drector a have ae obectve of axzng ther proft n akng the product deveopent pan, but a have dfferent contrant whch are baed on ther ndvdua condton. Obvou a BLMF decon probe occur coon n an organzatona decon practce, and nvove an dfferent decon tuaton whch are dependent on the reatonhp aong the foower. We have etabhed a fraework 4 for the BLMF decon probe, where nne an knd of reatonhp aongt the foower have been dentfed. The uncooperatve reatonhp, defned a the cae n whch there are no hared decon varabe aong the foower, the ot popuar one of BLMF decon probe n practce. Th uncooperatve reatonhp can ead to two tuaton. One that no foower take an reference fro other foower decon, reated 2, 5 reearch reut have been reported n terature. Anotheruncooperatve tuaton occur when depte the foower are uncooperatve n that there no harng of decon varabe, the do, however, cro reference nforaton b conderng other foower decon reut n each of ther own decon obectve and contrant. We ca th cae a a referentauncooperatve tuaton, and th paper w partcuar focu on th tuaton. We have deveoped an extended branch and bound agorth for ovng th probe. 6 Th paper further preent an extend kth-bet approach to ore effectve ove th probe. Th paper organzed a foow. In Secton 2, a ode for the referenta-uncooperatve tuaton of a near BLMF decon probe preented, and the defnton for an opta outon and reated theore are gven. An extended kth bet approach for ovng the referenta-uncooperatve BLMF decon probe propoed n Secton 3. A cae-baed exape for the extended kth-bet approach utrated n Secton 4. Concudng reark are gven n Secton A Mode for the Referenta-Uncooperatve BLMF Decon Probe A BLMF decon probe ha been defned to have two or ore foower at the ow ever of the beve probe. Under th defnton, f two foower don t have an hared decon varabe, t caed an uncooperatve reatonhp between the two foower. But f one of the ha a reference to another foower decon nforaton n h/her obectve or contrant, the two foower are defned a havng a referentauncooperatve reatonhp. When there a referentauncooperatve reatonhp n a BLMF decon ode, th ode caed a referenta-uncooperatve BLMF decon ode. We preent th ode a foow. n For x X R, Y R, F : X Y L Y R, and f : X Y Y R,,2, a near BLMF decon probe where ( 2) foower are nvoved and there are no hared decon varabe, but hared nforaton n obectve functon and contrant functon aong the foower whch defned a foow: n F(,, cx + d (a) x X ubect to ) Ax + B b ) Y ubect to A x + C b n n c R d R, p R, b R p n p R B q n R A R, C (b) n f,, c x + e (c) (d) where c R,, e, q b R, A,, q R,,,2,,. To fnd an opta outon for th ode (a)-(d), we ntroduce defnton of contrant regon, proecton of S onto the eader decon pace, feabe et for each foower, and nducbe regon for a near BLMF decon probe n Defnton. Pubhed b Atant Pre Coprght: the author 206

3 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), Referenta-uncooperatve Beve Mut-foower Decon Makng: Mode and the kth-bet Approach Defnton (a) Contrant regon of a near BLMF decon probe: S {,, ) X Y Y, Ax + k B b, A x + C b,,2, }. The contrant regon refer to a pobe cobnaton of choce that the eader and foower a ake. (b) Proecton of S onto the eader decon pace: C S( X ) { x X : Y, Ax + B A x + b, b,,2, } (c) Feabe et for each foower x S(X ) : S (x) { Y :, ) S}. The feabe regon for each foower affected b the eader choce of x, and the aowabe choce of each foower are the eeent of S. (d) Each foower ratona reacton et for x S(X ) : P ( x) { Y : arg n[ f ˆ, ),,2, : ˆ S ( x)]}. where,2, arg n[ f ˆ, ) : ˆ S ( x)] { S ( x) : f, ) f ˆ, ),,2,,, ˆ S ( x)} ( The foower oberve the eader acton and utaneou react b eectng fro ther feabe et to nze ther obectve functon. (e) Inducbe regon: IR {, ) : ) S, P ( x),,2, } Thu the ode gven b expreon (a)-(d) can be rewrtten n ter of the above notaton a foow n{ F(, ) : ) IR} (2) We propoe the foowng theore to characterze the condton under whch there an opta outon for a referenta-uncooperatve near BLMF decon probe hown n (a)-(d). Theore If S nonept and copact, there ext an opta outon for a near BLMF decon probe. Proof: Snce S nonept there ext a pont ( x,,, ) S. Then, we have x S( X) b Defnton (b). Conequent we have S x,,2, ( ) b Defnton (c). Becaue S copact and Defnton (d), we have P ( x ) { Y : arg n[ f ( x, ˆ, : ˆ S ( x )]} {,, ) f ( x, ˆ, ),,2, Y : { S ( x ) : f ( x,,,2, ), ˆ S( x )}} 0 ( x where,2,. Hence, there ext P ), 0 0,2, uch that ( x,, ) S. Therefore, we have: IR {, ) : ) S, P( x),,2,, } b Defnton (e). Becaue we are nzng a near functon n F(, cx + d over IR, x X ) whch nonept and bounded an opta outon to the near BLMF decon probe ut ext. 3. An Extended kth-bet Approach for the Referenta-Uncooperatve BLMF Decon Probe We frt gve a et of reated properte n th ecton. Baed on the et of properte an extended kth-bet approach for ovng referenta-uncooperatve decon probe preented. Theore 2 The nducbe regon can be wrtten equvaent a a pecewe near equat contrant copred of upportng hper pane of contrant regon S. Proof: Let u begn b wrtng the nducbe regon of Defnton (e) expct a foower: Pubhed b Atant Pre Coprght: the author 207

4 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), G. Zhang, C. Sh, and J. Lu IR {, ) : ) S, e n[ e : B b Ax B, Let u defne: ( C b A x, C,,2,, ],,2, } b T b b, b, ), A A, A, ), where,2, Now we have B ( B, C, ( T, C ) IR {, ) : ) S, e n[ e : B b A x B, ], Let u defne: Q, A, T,2, } (3) ) n[ e : B b A x, B ] where,2,,2,. For each vaue of x S(X ),, (4), the reutng feabe regon to probe () nonept and copact. Thu, for Q, whch a near progra paraeterzed n,,2, and, awa ha a outon. Fro duat theor we get ax{ u( A x + B b ) : ub e, u 0} (5), whch ha the ae opta vaue a (4) at the outon u. Let u, u be a tng of a the vertce of the contrant regon of (5) gven b U { u : ub e, u 0}. Becaue we know that a outon to (5) occur at a vertex of U, we get the equvaent probe:, ax{ u ( A x + B b ) : u { u, u }} whch deontrate that Q ) a pecewe near functon. Rewrtng IR a: ( IR {, k ) S : Q ) e 0 ( (6) ed dered reut.,2,,,2, } (7) Coroar The probe () equvaent to nzng F over a feabe regon copred of a pecewe near equat contrant. Proof: B (2) and Theore 2, we have the dered reut. Each functon Q defned b (4) convex and contnuou. In genera, becaue we are nzng a near functon cx + d over IR, and becaue F F bounded beow S b a n{ cx d : +, ) S }, the foowng can be concuded. Coroar 2 A outon for the near BLMF decon probe occur at a vertex of IR. Proof: A near BLMF decon probe can be wrtten a n (2). Snce cx + d near, f a outon F ext, one ut occur at a vertex of IR. Theore 3 The outon ( x,, ) of the near BLMF decon probe occur at a vertex of S. Proof: Let ( x,, ), ( x,, ) be the dtnct vertce of S. Snce an pont n S can be wrtten a convex cobnaton of thee vertce, et ( x,,, r ) r r r r α α ( x,, ), where, α 0,,2, r and r r. It ut be hown that r. To ee th et u wrte the contrant to () at ( x,,, ) n ther pecewe near for (7). 0 Q ) e Rewrte (8) a foow:,2,,,2,, (8) 0 Q ( α ( x, ) e ( α ) α Q ( x, ) α e where,2,,,2,, Pubhed b Atant Pre Coprght: the author 208

5 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), Referenta-uncooperatve Beve Mut-foower Decon Makng: Mode and the kth-bet Approach B convext of Q ), we have ( 0 α ( Q ( x, ) e where,2,,,2,,. But b the defnton, Q ( x, ) S ( x ) ) n e e,,2,,,2,,. Therefore, Q ( x, ) e 0,,2,, r,,2,,2,,.notng that α 0,, 2,, r, the equat n the precedng expreon ut hod or ee a contradcton woud reut n the equence above. Conequent Q ( x, ) e 0,,2, r,,2,,2,,. Th pe that ( x,,, ) IR,,2, r and ( x,,, ) can be wrtten a a convex cobnaton of pont n IR. Becaue ( x,,, ) a vertex of IR, a contradcton reut une r. Coroar 3 If x an extree pont of IR; t an extree pont of S. r r r Proof: Let ( x,, ), ( x,, ) be the dtnct vertce of S. Snce an pont n S can be wrtten a convex cobnaton of thee vertce, et ( x, r ) α ( x,, ), where α,, α 0,,2, r and r r. It ut be hown that r. To ee th et u wrte the contrant to () at ( x,,, ) n ther pecewe near for (7). 0 Q,,2,, ) e,2, Rewrte the above foruaton a foow: 0 Q ( α ( x, )) e ( α ) α Q ( x, ) α e where,2,,2,,. B convext of Q ), we have: ( 0 α ( Q ( x, ) e ) r where,2,,,2,. But b the defnton, Q ( x, ) S( x ) n e e,2,,2,, Therefore, Q ( x, ) e 0,,2, r,,2,,,2,. Notng that α 0,,2,, r, the equat n the precedng expreon ut hod or ee a contradcton woud reut n the equence above. Conequent Q ( x, ) e 0,,2, r,,2,,,2,. Th pe that ( x,,, ) IR,,2,, r and ( x,,, ) can be wrtten a a convex cobnaton of pont n IR. Becaue ( x,,, ) a vertex of IR, a contradcton reut une r. Th ean that ( x,,, ) an extree pont of S. Theore 3 and Coroar 3 have provded theoretca foundaton for a new agorth ued n our extended kth-bet approach. It ean that b earchng extree pont on the contrant regon S, we can effcent fnd an opta outon for a near BLMF decon probe. The bac dea of the agorth that accordng to the obectve functon of the upper eve, we arrange a the extree pont n S n a decendng order, and eect the frt extree pont to check f t on the nducbe regon IR. If e, the current extree pont the opta outon. Otherwe, the next one w be eected and checked. More pecfca et ( x,,, ),, ( N N x,, N, ) denote the N ordered extree pont to the near BLMF decon probe uch that: n{ cx + d :, ) S} (9) + + cx + d cx + d,,2,, N. Let (,,, 2 ) denote the opta outon to the foowng probe n{ f ( x,, ) : S ( x ),,2, } (0) Pubhed b Atant Pre Coprght: the author 209

6 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), G. Zhang, C. Sh, and J. Lu We on need to fnd the aet,2, N under whch,,2,. Let u wrte (0) a foow: n f(, ) ubect to S(x) x x where,2,. We on need to fnd the aet under whch,,2,. Fro Defnton (b), we rewrte (0) a foow: n f,, c x + e (a) ubect to ) Ax + B b A x + C (b) b,,2, (c) x x (d),,, 0, (e) 0 2 where,2,. Sovng th probe equvaent to eect one ordered extree pont ( x,, ) and then ove () to obtan the opta outon. If for a we have, then ( x,,, ) the goba optu to (a)-(d). Otherwe, check next extree pont. Baed on the reut obtaned fro above procedure, an extended kth-bet approach whch can ove a referenta-uncooperatve BLMF decon probe decrbed a foow. Step Put. Sove (9) wth the pex ethod to obtan an opta outon ( x,,, ). Let W ( x,, ) and T. Go to Step 2. Step 2 Sove () wth the bounded pex ethod. Let denote the opta outon to (). If for a,,, ( x,,, ) the goba optu to (a)-(d). Otherwe, go to Step 3. Step 3 Let W [] denote the et of adacent extree pont of x,, ) uch that ( x, ) W[ ] ( pe cx + d cx + d. Let T T {( x,, )} and W ( W W ]) / T. Go to Step 4. [ ( + Step 4 Set + and chooe x,,, ) o that cx + d n{ cx d : ( x,, ) W}. Go back to Step 2. The extended kth-bet approach ea to be ued to ove a near referenta-uncooperatve BLMF decon probe. 4. An Exape of Logtc Manageent Th ecton frt preent a ogtc pannng probe odeed a a referenta-uncooperatve BLMF decon probe. It then how how the propoed extended kthbet approach ued for ovng the probe. A ogtc chan often nvove a ere of unt uch a upper and dtrbutor. A the unt nvoved n the chan are nterreated n a wa that a decon ade at one unt affect the perforance of next unt(). In the eante, when one unt tre to optze t obectve, t a need to conder the obectve of next unt, and t decon w be affected b the next unt reacton a we. Both upper and dtrbutor, two portant unt n a ogtc chan, have ther own obectve uch a to axze ther beneft and nze ther cot; contrant uch a te, ocaton and facte; and varabe uch a prce. For each of pobe decon ade b the upper, the dtrbutor fnd a wa to optze h/her obectve vaue. The opta outon of the dtrbutor aow the upper to copute h/her obectve functon vaue. A the an purpoe of akng a ogtc pan to optze the upper obectve functon vaue, the upper the eader, and the dtrbutor the foower n the cae. We aue that there are two knd of dtrbutor A and B n th cae. The have ther own decon varabe, obectve and contrant. But the have cro reference of nforaton b conderng other foower decon reut n each of ther own decon obectve and contrant. For exape, dtrbutor A conder the prce of tranportaton of dtrbutor B. We therefore etabh a referenta-cooperatve BLMF ode for th probe. n For x X R the upper (eader ) decon varabe, Y R the dtrbutor A (foower A ) Pubhed b Atant Pre Coprght: the author 20

7 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), Referenta-uncooperatve Beve Mut-foower Decon Makng: Mode and the kth-bet Approach R decon varabe, z Z the dtrbutor B (foower B ) decon varabe, F : X Y R the upper obectve functon, and f : X Y R and f2 : X Y R the dtrbutor A and dtrbutor B obectve functon repectve. In order to ea how the ue for the propoed kth-bet approach, the ogtc pannng probe pfed nto X { x 0}, Y { }, Z { } wth x R, R, z R. The upper obectve to nze, over the et X, the tota tranportaton cot of the te decrbed b nf (. The dtrbutor A eek to nze h/her tranportaton te dea decrbed b n f over the et Y, and the dtrbutor B b n f 2 over the et Z. Athough the two knd of dtrbutor have dfferent decon varabe, decon obectve and contrant, but each of the take other decon varabe nto ther obectve and contrant a reference. Th a tpca referenta-uncooperatve BLMF decon probe. The probe ode preented a foow: n F( x z x X ubect to x z n f x + z Y ubect to x + + z n f 2 x + z z Z ubect to. Accordng to the extended kth-bet approach, th ode can be rewrtten n the forat of (9) a foow: n F( x z ubect to x z x + + z x 0. Now we go through th extended kth-bet approach fro Step to Step 4. In Step, et, and ove above probe wth the pex ethod to obtan an opta outon ( x [ ], [], z[] ) (2,0,0). Let W {(2,0,0)} and T φ. Go to Step 2. In the Loop : Settng and b (), we have: n f x + z ubect to x z x + + z x 2. Ung the bounded pex ethod, we have. Becaue of [ ], we go to Step 3 and then have W [ ] {(,0,0),(2,,0),(2,0,)}, T {(2,0,0)} and W { (,0,0), ( 2,,0),(2,0,)}. We then go to Step 4. Update 2, and chooe x,, z ) (,0,0), go back ( [ ] [ ] [ ] to Step 2. In the Loop 2: Settng and b (), we have n f x + z ubect to x z x + + z x. Sae a oop, b ung the bounded pex ethod, we have. Becaue of [ ], we go to Step 3, and obtan: W [ ] {(2,0,0),(,,0),(,0,)} T {(2,0,0),(,0,0)} W {(2,,0),(2,0,),(,,0),(,0,)} Pubhed b Atant Pre Coprght: the author 2

8 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), G. Zhang, C. Sh, and J. Lu Then go to Step 4. Update 3 x [ ], [ ], z[ ) ( ] (2,,0), then go to Step 2 agan. In the Loop3: Settng and we have b (): n f x + z ubect to x z x + + z x 2., and chooe Through ung the bounded pex ethod, we obtan and [ ]. Th a dfferent tuaton fro at oop. We thu et + and have a new expreon of dtrbutor functon f 2 b (): n f 2 x + z ubect to x z x + + z x 2. Sae a before, b ung the bounded pex ethod agan, we have z. Becaue z z[ ], we go to Step 3, and have: W [ ] {(2,0,0),(,,0),(2,,)} T {(2,0,0),(,0,0),(2,,0)} W {(2,0,),(,,0),(,0,),(2,,)}. We then go to Step 4. Updatng 4 and choong ( [ ] [ ] [ ] x,, z ) (2,0,), then we go back to Step 2. In Loop 4: Settng and we have b (): n f x + z ubect to x z x + + z x + + z x 2 8 Through ung the bounded pex ethod, we obtan and [ ]. We go to Step 3, have: W [ ] {(2,0,0),(,0,),(2,,)} T {(2,0,0),(,0,0),(2,,0),(2,0,)} W {(,,0),(,0,),(2,,)} We then go to Step 4. Updatng 5 ( ] x [ ], [ ], z[ ) (,,0) In Loop 5: Settng and we have b (): n f x + z ubect to x z x + + z x we get Through ung the bounded pex ethod, we obtan and [ ]. We et +, have: n f 2 x + z ubect to x z x + + z x We have: z, z z[ ], go to Step 3, we have: W [ ] {(,0,0),(,,),(2,,0)} T {(2,0,0), (,0,0), (2,,0), (2,0,), (,,0} W {(,0,),(2,,),(,,)} Pubhed b Atant Pre Coprght: the author 22

9 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), Referenta-uncooperatve Beve Mut-foower Decon Makng: Mode and the kth-bet Approach We then go to Step 4. Updatng 5 ( [ ] [ ] [ ] x,, z ) (,0,). In Loop 6: Settng and we have b (): n f x + z ubect to x z x + + z x we get Through ung the bounded pex ethod, we obtan and [ ]. We go to Step 3, have: W [ ] {(,0,0), (,,),(2,0,)} T {(2,0,0),(,0,0),(2,,0),(2,0,),(,,0),(,0,)} W {(2,,),(,,)} We then go to Step 7. Updatng 5 get x,, z ) (2,, ) ( [ ] [ ] [ ] In Loop 7: Settng and we have b (): n f x + z ubect to x z x + + z x 2 we Through ung the bounded pex ethod, we obtan and [ ]. We et +, have: n f 2 x + z ubect to x z x + + z x 2 We have: z, z z[ ]. Go to Step 4, we have: ( ] x [ ], [ ], z[ ) (2,,). It ha been found that fro oop 7 that the opta outon of the referenta-uncooperatve BLMF probe occur at the pont ( x,, z ) (2,, ) wth the eader obectve vaue F 3 2, and two foower obectve vaue f 2 and f 2 repectve. 5. Concuon and Further Stud A referenta-uncooperatve BLMF decon probe occur coon n anageent and pannng of an organzaton. For ovng uch a BLMF decon probe, th paper extended the kth-bet approach fro deang wth pe one-eader-and-one-foower tuaton to copex referenta-uncooperatve utpe foower tuaton. Th paper further utrated the deta of the propoed approach b an exape of ogtc pannng probe. Inta experent reut howed that th extended kth-bet approach can effectve ove the propoed BLMF decon probe. Soe practca ue of th extended approach w be condered a our future reearch tak for BLMF decon akng. Acknowedgeent The work preented n th paper wa upported b Autraan Reearch Counc (ARC) under dcover grant DP Reference. J. Bard, Practca beve optzaton: agorth and appcaton (uwer Acadec Pubher, USA, 998). 2. J. Bard and J. Fak, An expct outon to the ut-eve prograng probe, Coputer and Operaton Reearch, 9(982) J. Bard and J. Moore, A branch and bound agorth for the beve prograng probe, SIAM Journa of Scentfc and Stattca Coputng, (990) W. Baa and M. arwan, On two-eve optzaton, IEEE Tran. Autoatc Contro, AC-27 (982) P. Hanen, B. Jauard, and G. Savard, New branch-and-bound rue for near beve prograng, Journa on Scentfc and Stattca Coputng 3 (992) Pubhed b Atant Pre Coprght: the author 23

10 Internatona Journa of Coputatona Integence Ste, Vo., No. 3 (Augut, 2008), G. Zhang, C. Sh, and J. Lu 6. D. Whte and G. Anandanga, A penat functon approach for ovng b-eve near progra, Journa of Goba Optzaton 3 (993) W. Cander and R. Towne A near two-eve prograng probe, Coputer and Operaton Reearch 9 (982) J. Bard, An nvetgaton of the near three eve prograng probe, IEEE Tranacton on Ste, Man, and Cbernetc 4 (984) W. Baa and M. arwan, Two-eve near prograng, Manageent Scence 30 (984) G. Savard, Contrbuton á a prograaton athéatque á deux nveau PhD the (Unverté de Montréa, Écoe Potechnque, 989).. C. Sh, J, Lu and G. Zhang, An extended th-bet approach for near beve prograng, Apped Matheatc and Coputaton 64 (2005) C, Sh, G. Zhang and J. Lu, On the defnton of near beve prograng outon, Apped Matheatc and Coputaton 60 (2005) C. Sh, G. Zhang and J. Lu, A th-bat approach for near beve ut-foower prograng, Journa of Goba Optzaton 33 (4) (2005) J. Lu, C. Sh and G. Zhang, On beve ut-foower decon-akng: genera fraework and outon, Inforaton Scence 76 (2006) J. Lu, C. Sh, G. Zhang and D. Ruan, Mut-foower Lnear Beve Prograng: Mode and uhn-tucker Approach, Apped Coputatona Integence, n Proceedng of the IADIS Internatona Conference n Apped Coputng (Portuga, Agarve, 2005), pp J. Lu, C. Sh, G. Zhang and D. Ruan, An extended branch and bound agorth for beve ut-foower decon akng n a referenta-uncooperatve tuaton, Internatona Journa of Inforaton Technoog and Decon Makng 6 (2) (2007) Pubhed b Atant Pre Coprght: the author 24