Online Supplement to Reactive Tabu Search in a Team-Learning Problem

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1 Olie Suppleme o Reacive abu Search i a eam-learig Problem Yueli She School of Ieraioal Busiess Admiisraio, Shaghai Uiversiy of Fiace ad Ecoomics, Shaghai 00433, People s Republic of Chia, she.yueli@mail.shufe.edu.c INFORMS Joural o Compuig. Liearizaio of Problem P Due o he symmeric propery of (), we ca recas Problem P io: Problem P Mi Z = 4 ' < ' k < W m W k < m, m k > m Subjec o W + Wmk = L W W = W W, k A umber of papers i he lieraure have ried o liearize quadraic ieger programmig problems (Oral ad Keai 99). We adop a sadard approach proposed by Waers (967) by iroducig he followig biary variables: Y ' = W W, < ', k m; X = W W, k < m l; V = W W, k < m l. ' < l < l < Wih hese defiiios, we eed o solve a liear ieger program equivale o Problem P : Problem IP Mi Z = 4 ' < ' k < Y m k < m, m k > m Subjec o W + Wmk = L X = V, k l l ' ' W + W Y < ', k < m ' ' W W + Y 0 < ', k < m

2 W + W X, k l W W + X 0, k l W + W V, k l W W + V 0, k l Cosiderig {... }, k, m, l {... K} where K =, here are K ( K )(4K + 3 5) K( K ) biary variables ad K + (5K + 3 3) cosrais i Problem IP. For a case 6 (, S, L) = (5,4,4), his correspods o abou 7,00 biary variables ad 6,000 cosrais, which is formidable for curre IP solvers.. Proof of Proposiio I he firs erm, all he cohors ca be arbirarily assiged, such as (,,...,L)(L+,L+,...L)...((S-)L+,(S-)L+,...) i our early oaios. If we igore he issue of balace, we fix oe cohor, such as cohor, o fid he maximum exposure rae i ca achieve. If S, i he oher - erms, he secios wih cohor ca be assiged as (,L+,L+,...L-), (,L+,L+,...,3L-)...(,(-)L+, (-)L+,...,L-). By such a assigme, all he cohors ha cohor mees are differe, ad he cohor has me he maximum umber of oher cohors, which is (L-). Accordig o he defiiio, i his case, cohor reaches he maximum exposure rae ( L ) +. Cosiderig ha, ormally, >, we simply have ( L ) + = S his complees he proof. + < S 3. Proof of Proposiio his proposiio ca be proved via wo approaches.

3 Firs, if we have a perfec imeable wih parameers (, S, L), accordig o he defiiio, he oal umber of pairig is C. O he oher had, from he imeable cofiguraio, his umber ca be coued by he pairig umber i a erm muliplied by he erm umber, which is L L SC. herefore we have he relaioc SC irisic approach ad is model-idepede. =, ad (0) i he paper follows. his is a Secod, if we have a perfec imeable, boh Z 0 ad Z should be zero. herefore, LB mus be zero, from which we ca also derive he resul of Proposiio. 4. Proof of Corollary Proof: By reaig α cohors as a sigle cohor cluser, we ca allocae he clusers accordig o he perfec imeablig (, S, L), ad he we arrive a a fully-exposed imeable. 5. Proof of Proposiio 3 Several facs lead o his coclusio,. Permuaios i differe erms are idepede.. he umber of permuaios of elemes is!. 3. Neiher he permuaios of he cohors i he same secio or he re-orderig of he secios will chage he imeable. 6. Proof of heorem o prove heorem, we eed o review he coceps of group heory. 6.. Elemeary Group heory A group is a se of elemes G={A,B,C,...} wih muliplicaio ad iverse operaios over hese group elemes. A group saisfies he followig four axioms: () I G, such ha A G, AI = IA= A; I is called he ideiy eleme. () A G, A G, such ha AA = A A =I (3) AB, G, C G, such ha AB = C 3

4 (4) ABC,, G, ( AB) C = A( BC) From ()-(4), group operaios are he same as marix operaios, such as( AB) = B A. Oher ermiologies i group heory used i his paper are (a) sub-group: a subse g G, such ha g is also a group. (b) group geeraor: a subse g G, such ha all he elemes i G ca be geeraed from g by muliplicaio ad iverse operaios. 6.. Permuaio Group A permuaio group is a special group. Cosiderig a collecio of elemes i he order (,..), all possible operaios o chage he order of hese elemes build up he permuaio group P. A well-kow resul for he permuaio of elemes is ha he dimesio of!. he simples eleme i P is he pair-wise exchage P ij ha permues he pair (i, j). wo impora properies of permuaio group used i his Olie Suppleme are P is (P) P P ij = ij (P) All he exchages P ij i P are he geeraors of P, i.e., ay permuaio of he elemes ca be wrie as fiie muliplicaios of hose pair-wise exchages. For elemes, here are C such exchages Proof of heorem Sice permuaios i differe erms have o effec o each oher, he chage from oe cofiguraio of he imeable o aoher ca be decomposed io idividual permuaios i each erm. For oe erm wih S secios ad L cohors i each secio, we deoe 0 =(,,...L)(L+,L+,...L)...((S-)L+,(S-)L+,..., ) as he sarig cofiguraio. I is clear ha exchagig cohors i he same secio or he reorderig of he secios do o chage he imeable cofiguraio. While he permuaios amog a secio are excluded i he algorihm i he paper, hey are icluded i his proof. he reaso is ha he orivial permuaios cao cosiue a group by hemselves, which ca be illusraed by a example. Suppose we have hree cohors (k, m, l), he a basic propery kow o permuaio groups is P P P = P. If cohors 4

5 k ad m are i he same secio bu differe from cohor l, he operaio o he lef side of his ideiy is composed of hree o-rivial permuaios; however, he righ side is a rivial operaio ha does o chage he cofiguraio. I oher words, we mus keep hose rivial permuaios i order o keep he closure of he group operaios. Now ay possible cofiguraio i a erm is a re-orderig of hese elemes, hough permuaios amog he same secio are rivial. ha is o say, for a ew imeable cofiguraio, here exiss a such ha = 0. If we deoe all he pair-wise exchages as {,,... ( ) }, we have i geeral = i j... k, where i, j..., k {,,... ( ) }, ad he umber of he pair-wise exchages is fiie. For a oe-erm cofiguraio, if he real opimal soluio has he cofiguraio op ad he iiial soluio has he cofiguraio ii, here exis permuaios op ad ii such haop = op0, ii = ii0, where 0 is defied i he begiig of his proof. Due o he iverse operaio of group operaios, we have ad ii =. From (P), we ca wrie op op op ii ii as he producs of pair-wise exchages of,,... ( ), Hece =..., =... op i j k ii = (... )(... ) α β γ op i j k α β γ ii = ( i j... k)( γ... β α ) ii = (... )(... ) i j k γ β α ii where he group operaio rule, ( AB) = B A, ad for he exchage operaor propery, i =, are used i he equaliies. i he fial sep says ha, for a give erm, here exiss a fiie umber of pair-wise exchages ha lead o he opimal soluio from ay iiial feasible soluio. Sice he operaios i all he erms are idepede, i is also rue for he whole imeable. hus we complee he proof for heorem. 5

6 7. Improved Hill Climbig Heurisic he IHCS heurisic is saed as: Sep Iiialize a feasible soluio, he derive he decisio variables W[][k][m] Objecive:= Z value of he iiial soluio from () Sep Searchig procedure. Ipu cou_max // maximal ieraios whe soluio is degeerae cou:=0 idex:= // idicae if more moves are eeded. While idex= ad Z > Z _ lower _ boud, do Fid if here is a pair-wise exchage ha ca improve he curre soluio or move o a geeraed soluio. If here is o such a move, se idex=0 If idex=, he updae he bes soluio by exchagig his pair If his pair-wise exchage does o improve Z, he cou=cou+ // cou he ime of degeeracy Else cou=0 // he soluio is improved EdIf If cou=cou_max, he idex=0 // sop search if umber of degeeraed ieraios is up o cou_max EdIf EdWhile Sep3 If he bes soluio foud is equal o he lower boud, he soluio is opimal Derive he imeable from W[][k][m] Some explaaios o IHCS: Sep. We guess ha he fial soluio should deped o he iiial feasible soluio. However, i he experimes, we will choose he wors iiial soluio where he cofiguraios i all he erms are he same. he reaso for such a choice is ha we ca sysemaically geerae he iiial feasible soluio for ay parameers i codig. Moreover, he firs erm i he imeable is always fixed durig he search sice i is always free o choose he firs erm. 6

7 Sep. his is he mai procedure o search for he bes soluio, where we se wo codiios o sop. he firs is corolled by he lower boud of Z, if here is such a value. I he compuaioal experimes, we have se his lower boud as 0. he secod is he codiio by a corol parameer cou_max ha meas he maximal ieraios o search i he eire eighborhood sice he objecive value is las improved. I fac, he secod is useful whe he searchig process mees degeeraed soluios. cou_max is se o 0 for all he cases esed i Secio 6. Ay of hese wo codiios will discoiue he search process ad provide he bes soluio. Refereces Oral, M., O. Keai. 99. A liearizaio procedure for quadraic ad cubic mixed-ieger problems. Operaios Research 40 S09-6. Waers, L Reducio of ieger polyomial programmig problems o zero-oe liear programmig problems. Operaios Research