A General Column Generation Algorithm Applied to System Reliability Optimization Problems

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1 A Genera Coumn Generaton Agorthm Apped to System Reabty Optmzaton Probems Lea Za, Davd W. Cot, Department of Industra and Systems Engneerng, Rutgers Unversty, Pscataway, J 08854, USA Abstract A genera coumn generaton approach for system reabty optmzaton s descrbed and demonstrated. In prevous years, a tremendous amount of study has been concentrated on system reabty optmzaton and coumn generaton as a technque to optmze arge scae probems. Ths paper can be consdered as a contnuaton and ntegraton of these two topcs. It presents new deas for formuatng the aocaton probems n reabty systems and opens a new area for appcaton of coumn generaton. We present the reformuaton of probems ncudng the redundancy aocaton probem, reabty aocaton probem and redundancy-reabty aocaton probem. Then, we descrbe how the coumn generaton agorthm can be used to sove these probems. Ths approach obtans the best found soutons for eampe probems whe requrng a sma fracton of computatona efforts requred by other methods. Key Words Coumn generaton agorthm, reabty optmzaton, redundancy aocaton probem, redundancyreabty aocaton probem. Introducton Coumn generaton (CG can be consdered as one of the most successfu approaches for sovng arge-scae nteger programmng probems over the ast decade. The dea was ntroduced by Gmore & Gomory [ 0] when fndng the optma souton for the cuttng-stoc probem. The probem was to cut umber whch were n dfferent engths to meet demands for specfc engths of sheves. They notced that, t s practcay mpossbe to generate a the possbe confguratons of umber cuts. Therefore, they devsed a methodoogy whch started wth a set of feasbe confguratons, and used the optma souton to the smpfed probem to generate a new promsng confguraton. Later, the dea has been used by Mnou [ ] as a powerfu technque to reformuate some mportant combnatora probems. Vance et. a. (994 combned CG and branch-and-bound to present an agorthm for bnary cuttng stoc probems. From the appcaton pont-ofvew, Desrosers et. a. [ 3] apped CG to routng/dstrbuton probems; Ryan [ 4] used CG n arcrew rosterng and Vanderbec [ 5] suggested eact effcent agorthm based on CG for the cuttng stoc probem. There are three varatons of CG accordng to Whem [ 6]. The common dea s to defne a Master Probem (MP whch s the subset of the orgna probem. The MP contans ony a seected number of coumns. In Type I CG, the MP nteracts ony once wth another probem, caed the Auary Probem (AP whch sends the attractve coumns to the MP. Optmzaton of the master probem s based on the coumns sent from AP. In Type II CG, the nteracton of the MP s wth another probem, caed the Sub-Probem (SP or prce-out probem. Type III CG s based on Dantzg-Wofe decomposton (Dantzg et. a. [ 7] n whch the MP teratvey nteracts wth one or more SPs to dentfy promsng coumns. The CG whch s consdered n ths paper s of Type II. The probem has one MP and one (or more SP(s to prce-out the MP usng the smpe optmaty condton. Ths means that the probem starts wth a MP whch s a subset of the orgna probem wth ony a subset of the coumns. The optmzaton of the MP s based on avaabe methods for optmzaton. Then, SP(s are constructed wth the dea of smpe optmaty whch s to consder a non-basc varabes (coumns. The nteracton between the MP and the SP(s can be demonstrated as n Fgure. Ths fgure shows the decomposton of the probem nto two probems: Master Probem and Sub-Probem(s. As t s shown, the dua varabes from the MP are sent to the SP(s, and n return, each SP, wth postve objectve functon, sends ts coumn to the MP.

2 Master Probem Dua Varabes Coumn(s Sub-Probem(s otaton and Abbrevatons otaton r M ( R r, r2,, r Fgure : Decomposton of the Probem nto MP and SP reabty of component number of components/subsystems n the system number of dfferent resource restrctons reabty of the system as a functon of component reabtes ( R, 2,, (,,, g r r2 r b r, r u reabty of the system as a functon of number of each redundant component type consumpton of resource as a functon of component reabtes amount of resource ower bound and upper bound on reabty of component j, u j ower and upper bound for number of components type j n subsystem t number of component choces for subsystem n number of master probem coumns consdered for subsystem (, 2,, t number of component type j used n subsystem j th component seecton vector for subsystem number of component type j used n subsystem for the j vector R ( reabty of subsystem λ ω α th, f component seecton vector s chosen for subsystem 0, otherwse vector of dua varabes reated to cost and weght constrants dua varabe reated to the th convety constrant th component seecton Abbrevatons RAP MP SP CG Redundancy Aocaton Probem Master Probem Sub-Probem Coumn Generaton

3 TS GA ACO Tabu Search Genetc Agorthm Ant Coony Optmzaton 2. Reabty Optmzaton Modes and Coumn Generaton The goa n any reabty optmzaton mode s to mamze the reabty of a gven system, often by consderng enhancement of component reabty, use of redundant components, or both. Three dfferent reabty optmzaton probems can be defned, dependng on the opton n hand to ncrease the system reabty. The frst probem s caed reabty optmzaton probem. In ths case, the ony varabe s the reabty of each component whch s a contnuous varabe. Ths probem can be represented as: ( ma R r, r2,, r { r, r2,, r u g r, r,, r b : p,2,, m ; r r r : j,2,, ( { { p 2 p j j j The second s caed redundancy aocaton probem (RAP. Here, to ncrease the reabty of the system one can add redundant components to the nta components. The varabes are the number of components to be assgned to each subsystem. The genera form of ths probem s: ( ma R, 2,, {,,, 2 + (,,, : {,2,, ; : {,2,,, {,2,, g p 2 bp p m j j t The thrd s the reabty-redundancy probem. Here, the decson has to be made on how many components are necessary from each component choce and wth what reabty. The genera form s as foows: ( ma R r, r2,, r;, 2,, { r,, r ;,, g r, r,, r ;,,, b : p,2,, m ; u :,2,,, j,2,, t ( 2 2 { { { u + : {, 2,, ; : {, 2,,, {, 2,, p p j j j rj rj rj j j j t Each of the three types of probems descrbed above has a nonnear objectve functon. The constrant(s can be near or nonnear, dependng on the probem nstance. The reabty optmzaton probem s a nonnear programmng probem wth contnuous varabes. There are aready dfferent souton technques for nonnear optmzaton whch offer very good resuts for the precson desred. The ast two probems are more compe. ot ony are the probems nonnear, but aso, the compety of the probems hghy ncreased wth nteger varabes. That s the reason we consder the ast two probems for the appcaton of CG. To appy CG to each mode, we defne a MP and one (or mutpe SP(s, dependng on the probem. 2.. Redundancy Aocaton Probem The frst nstance to be consdered s RAP. We show the appcaton of CG to a seres-parae system wth subsystems n seres because, the objectve functon can not be demonstrated n genera for a networ wth unspecfed structure. The mode of such a probem can be demonstrated as:

4 t ma ( ( r,..., j = j= j + (,,, : {,2,, ; Z : {,2,,, {,2,, g b p m j t p 2 p j In the above mode, t s the number of dfferent component types that can be assgned to subsystem. Tang the natura ogarthm of the objectve functon, convertng the mamzaton probem to a mnmzaton one, and assumng separabty for g p,.e. g (,,, g ( MP to: Master Probem n t mn λn ( rj = = j = n λ g b : p,2,, m ; λ = : {,2,, ; p p = = = { λ 0, : {, 2,,, {, 2,, n j n ( { p 2 p = =, transforms the n whch, λ =, f the th component seecton vector s chosen for subsystem, 0 otherwse and n s the number of the MP coumns consdered for subsystem. ow, by ntazng the startng coumns, the probem s a near reaed probem n whch λ s are the varabes. Consderng the smpe optmaty condton, the SPs can be defned for each subsystem as: Subprobem m t ma j ωpgp( + α + n( ( rj p= j= + Z :,2,,, j,2,, t j { { To sove the SP(s, we use the dea presented by Barnhart et. a. [ 8] whch s, a good souton of the SP s suffcent to be sent to the MP. Therefore, we appy a heurstc to fnd a good souton for ths nonnear nteger programmng probem. Then, for each subprobem wth nonnegatve objectve functon, the vector s sent to the MP and the MP s reoptmzed. The process contnues unt there s no subprobem wth nonnegatve objectve functon. As the fna step, the MP s optmzed usng one of the avaabe technques for nteger programmng, for nstance branch-and-bound. The vaue for system reabty and component choces can be found from ths method Reabty-Redundancy Probem The same dea can be apped to reabty-redundancy probem. The resuts can be obtaned usng an anaogous procedure. The ony dfference s that n ths case, we have n subprobems, one for each component (nstead of each subsystem and each SP has ony two varabes: number of redundant components and the reabty of each component type.

5 As an eampe, consder a seres system wth components. For each component, there are choces to change the reabty and the number of redundant component. Assumng separabty for each resource consumpton constrant, (.e. g p(, 2,, ; r, r2,, r = gp(, r, the MP s: Master Probem n m ma λn ( r = = j = n λ λ { p p = = = = g, r b : p,2,, m ; λ = : {,2,, ; 0, : {, 2,,, {, 2,, n n ( { The subprobem for each component s: Subprobem m t mn ωpgp( + α + n( ( r p= j= + u Z :,2,, ; r r r : j,2,, { { j j j otce that each SP n ths case contans ony two varabes, whch shows the th component seecton for component, and r whch s the reabty of component. 3. Eampe As an eampe for our approach, we appy CG to the famous probem n RAP descrbed by Fyffe et. a (968. For ths probem, there are 4 subsystems for whch we can choose from three or four component choces, wth specfed reabty, cost and weght. There are weght and cost constrants. The component cost, weght and reabty data were orgnay presented by Fyffe et. a. [ 9]. The nstance consdered n ths eampe s one of the 33 varatons of the probem, as devsed by aagawa and Myaza [ 9]. The probem s demonstrated as: Master Probem 4 n m ma λn ( rj = = j = 4 n t 4 n t n j λ λ λ = λ { c C; w W;, S; 0,, S, {,2,, n j j j j = = j= = = j= = Subprobem(s t t t ma ω c 2 ( ( j j j + ω wj j + α + n rj j= j= j= + Z :,2,,, j,2,, t j { {

6 For the case of W=90 and C=30, the resuts can be summarzed n the foowng tabe. In ths tabe, the resut of CG s aso compared wth the resuts of prevous approaches for ths probem. As t s shown n the tabe, CG and TS gve best soutons for ths nstance. It s mportant to notce that CG resut s as good as TS, athough t ony consders around seventy coumns whe TS searches appromatey 20,000 soutons. Tabe : Comparson of Coumn Generaton resut wth Best Avaabe Ones GA TS ACO [ ] [ 2] [ 3] CG Reabty Reabty Reabty Reabty References. Gmore, P. C., and Gomory, R. E., 96, a near programmng approach to the cuttng stoc probem, Operatons Research, vo. 9, pp Mnou, M. 987, A cass of combnatora optmzaton probems wth poynomay sovabe arge scae set-coverng/ parttonng reaatons, RAIRO 2, Desrosers, J., Dumas, Y., Soomon, M. M., and Soums F., 990, Tme constraned routng and schedung. In: M.E. Ba, T.L. Magnant, C. Monna and G.L. emhauser, Edtors, Handboos n Operatons Research and Management Scence: etwors, orth-hoand, Amsterdam, Ch.. 4. Ryan, D. M. 992, The souton of massve generazed set parttonng probems n arcrew rosterng, vo. 43, no. 5, pp Vanderbec, F., and Wosey, L. A., 996, An eact agorthm for IP coumn generaton, Operatons Research Letters, vo. 9, pp Whem, W. E., 200, A technca revew of coumn generaton n nteger programmng, Optmzaton and Engneerng, vo. 2, pp Dantzg, G. B., and Wofe, P., 960, Decomposton prncpe for near programs, Operatons Research, vo. 8, pp. 0, Barnhart, C., Johnson, E. L., emhauser, G. L., Savesbergh, M.W. P., and Vance, P. H., 998, Branch and prce: Coumn generaton for sovng huge nteger programs, Operatons Research, vo. 46, no. 3, pp Fyffe, D. E., Hnes, W. W., and Lee,. K., 968, System reabty aocaton and a computatona agorthm, IEEE Transactons on Reabty, vo. R-7, no. 2, pp aagawa, Y., and Myaza, S.,98, Surrogate constrants agorthm for reabty, optmzaton probems wth two constrants, IEEE Transactons on Reabty, vo. R-30, no. 2, pp Cot, D. W., and Smth, A. E., 996, Reabty optmzaton of seres-parae systems usng a genetc agorthm, IEEE Transactons on Reabty, vo. 45, no. 2, pp Kuture-Kona, S., Smth, A. E., and Cot, D. W., 2003, Effcenty Sovng the Redundancy Aocaton Probem Usng Tabu Search, IIE Transactons, vo. 35, no. 6, pp Lang, Y. C. and Smth, A. E., 2004, An Ant Coony Optmzaton Agorthm for the Redundancy Aocaton Probem (RAP, IEEE Transactons on Reabty, vo. 53, o. 3, pp