IIE Transactons (00) 34, 1087 1100 Sequental DOE va dynamc programmng IRAD BEN-GAL 1 and MICHAEL CARAMANIS 1 Department of Industral Engneerng, Tel Avv Unversty, Ramat Avv, Tel Avv 69978, Israel E-mal: bengal@eng.tau.ac.l Department of Manufacturng Engneerng, Boston Unversty, 15 St. Mary s Street, Broolne, MA 0446, USA E-mal: mcaraman@bu.edu Receved July 000 and accepted May 00 The paper consders a sequental Desgn Of Experments (DOE) scheme. Our objectve s to maxmze both nformaton and economc measures over a feasble set of experments. Optmal DOE strateges are developed by ntroducng nformaton crtera based on measures adopted from nformaton theory. The evoluton of acqured nformaton along varous stages of expermentaton s analyzed for lnear models wth a Gaussan nose term. We show that for partcular cases, although the amount of nformaton s unbounded, the desred rate of acqurng nformaton decreases wth the number of experments. Ths observaton mples that at a certan pont n tme t s no longer effcent to contnue expermentng. Accordngly, we nvestgate methods of stochastc dynamc programmng under mperfect state nformaton as approprate means to obtan optmal expermentaton polces. We propose cost-to-go functons that model the trade-off between the cost of addtonal experments and the beneft of ncremental nformaton. We formulate a general stochastc dynamc programmng framewor for desgn of experments and llustrate t by analytc and numercal mplementaton examples. 1. Introducton and lterature revew Desgn Of Experments (DOE) s appled to help an expermenter gan nformaton about a partcular process or system through experments. DOE and n partcular Response Surface Methodology (RSM) comprse a group of statstcal technques for emprcal model buldng and analyss, seeng to relate a response Y to the values of control factors n 1 ; n ;...; nn (Myers and Montgomery, 1995). In some systems the nature of the relatonshp between Y and the n 0 s s nown exactly on the bass of underlyng engneerng, chemcal, or physcal prncples. In many cases, however, the underlyng physcs are not fully understood and the expermenter must approxmate the unnown response functon gðþ by an emprcal model: Y ¼ ^g n 1 ; n ;...; n n ; B 1 ; B ;...; B p þ e; ð1þ where, n most cases, ^g s a frst-order or a second-order polynomal; B 1 ; B ;...; B p are the parameters, whch the expermenter needs to estmate; and e s an addtve nose component. In practce, estmators are often obtaned by the method of least mean squares or maxmum lelhood from a set of m experments. Experments are represented by the m p desgn matrx X, whose jth row corresponds to the jth experment and columns are assocated wth polynomal terms (random varables are denoted here by captal letters except for the desgn matrx; vectors and matrces are bolded). Thus, the ðj; Þ entry of X ðj ¼ 1;...; m; ¼ 1;...; pþ reflects the level of a specfc factor or the nteracton of two or more factors n the jth experment. An mportant dstncton s often made between expermentaton procedures that see to maxmze (or mnmze) the value of the expermental response, and those procedures amng to gan nformaton about a system. A good example for the frst class of methods s the RSM whose objectve s to move sequentally to more promsng desgn regons wth respect to a predefned response objectve and to refne gradually the emprcal model. Lndley (1956) proposed the maxmzaton of the expected Shannon nformaton as a desgn crteron belongng to the second class of methods where the objectve of expermentaton s not to reach a decson but rather to gan nowledge about the world. Bernardo (1979) adopted ths crteron and showed that t s a specal case of the normatve DOE procedure to select a utlty functon, assess the probabltes, and to choose that desgn of maxmum expected utlty. In partcular, he maxmzes the expected nformaton n a desgn wth 0740-817X Ó 00 IIE
1088 Ben-Gal and Caramans respect to the posteror Bayesan densty of a quantty of nterest. The posteror densty captures personal opnons of the scentsts about the quantty after the experment s performed. The author clearly demonstrated how the expected nformaton crteron arses naturally from a decson problem of statstcal nference wthn a Bayesan framewor. DeGroot (196) generalzed Lndley s approach by suggestng a unversal uncertanty and nformaton functon, whch s not necessarly based on Shannon s entropy functon. Another mportant dstncton s made between the comprehensve (off-lne) approach and the sequental (on-lne) approach to DOE. Extensve lterature has been devoted to off-lne selecton of desgn matrces (Box and Draper, 1971, 1987; Atnson and Donev, 199) and offlne optmzaton of desgns wth respect to varancereducton related crtera (St. John and Draper, 1975; Lucas, 1976). Most publcatons consder the number of experments and the number of levels, that each control factor may be set to, as fxed constrants of the desgn problem. Thus, for a gven number of experments and factor levels, one tres to determne the best desgns relatve to specfed performance crtera, such as maxmum resoluton and mnmum predcton varance. A consderably smaller lterature addresses desgn costs and revenues by usng an on-lne sequental approach, where, after each experment s completed, the accumulated nformaton s used to specfy the next desgn. Box et al. (1978) clamed that the worse tme to desgn an experment s at the begnnng, when the expermenter nows the least. As a general recommendaton they proposed the 5% rule of thumb, accordng to whch not more than one-quarter of the expermental budget should be used n the frst desgn. In ths paper we develop a formal optmzaton approach for the sequental desgn problem that ncludes an nformaton-theoretc measure not consdered by Box et al. (1978). In the Sequental Hypothess Testng (SHT) problem (Bertseas, 1995) the desgner s nterested n selectng one of two hypotheses. At tme, after observng Y 0 ; Y 1 ;...; Y he has to decde whether to mae an addtonal observaton at cost c > 0, or to stop expermentng and accept a hypothess wth a hgher probablty of error. In contrast to the approach proposed n ths paper, the ncurred costs n the SHT are related to the probablty of mang an erroneous selecton and not to the value of the nformaton obtaned. Moreover, n SHT the observaton space s assumed to be fnte. Hardwc and Stout (1995) consdered an on-lne expermentaton nvolvng two Bernoull populatons. They suggested an algorthm for desgn of optmal experments n whch adaptve samplng s performed n stages. However, unle our approach, the total sample sze for the experment was consdered to be fxed. Krehbel and Anderson (199) proposed a monetary loss functon to determne the optmal fractonal replcate of a fractonal experment. The loss functon ncorporates the cost of producng estmates wth larger varances and the cost assocated wth expermentaton. We propose a smlar cost functon, snce, n certan cases, the reducton n the varance of the estmates s proportonal to the ncrease n Shannon nformaton expected from the experment. However, our cost functon s not lmted to fractonal experments. Box and Hunter (1965) consdered the problem of sequental constructon of D-optmal desgns for nonlnear models. Ther wor employs a Bayesan paradgm and lnear system approxmatons. At every expermentaton stage they add the tral that maxmzes the determnant of the nformaton matrx, whch, under certan assumptons, s equvalent to Shannon s nformaton measure. However, they dd not consder the explct cost of experments. The forward procedure that they use for the sequental constructon of varanceoptmal desgns s smlar to the coordnate exchange algorthm proposed by Feodorov (197) and used by the lmted loo-ahead approach employed as a buldng bloc of the optmal DOE algorthm developed here. Berlner (1987) adopted a smlar Bayesan approach to control the output of a mxture lnear model by choosng adequate values of ndependent varables usng quadratc programmng. Box (199) mentoned several strateges by whch a second stage of expermentaton mght evolve as a result of the analyss of the frst stage. Hs wor, however, does not address ssues of expermentaton cost. Ben-Gal et al. (1999) suggested a probablstc sequental methodology (PSM) for desgnng a factoral system, whch s based on sequental expermentaton, statstcal nferences and a probablstc local search. However, the expermentaton costs are agan not modeled drectly n that wor. Sequental expermentaton was dscussed by DeGroot (196) n relaton to the optmal selecton of experments when the goal s ether to mnmze the expected uncertanty after a fxed number of experments or to mnmze the expected number of experments needed to reduce uncertanty to a fxed level. Bradt and Karln (1956) formulated a dynamc programmng soluton approach for partcular problems of ths nature. Followng Lndley (1956) and Bernardo (1979) we use Shannon s entropy functon as a measure of uncertanty and am to reduce ts expected value through expermentaton. It s shown that nformaton theory measures motvate the use of a stochastc dynamc programmng approach for DOE, and, for lnear Gaussan models, are related to the well-nown D-optmalty crteron. Accordngly, we follow the wor of Bradt and Karln (1956) and DeGroot (196) to develop a Dynamc Programmng (DP) approach for sequental expermentaton. However, n our DP formulaton we also consder the expermentaton cost n addton to the gan from uncertanty reducton. The proposed framewor optmzes both the number of sets of experments as well as the actual desgn of each set of experments. The number of experments n
Sequental DOE va dynamc programmng 1089 each set s optmzed by provng and relyng on the fact that the expected ncremental nformaton ganed from addtonal experments decreases monotoncally wth the number of experments. The desgn of the next set of experments s optmzed at each step of the algorthm wth respect to the trade-off between the cost of an addtonal experment and the benefts of the expected ncremental nformaton. We present and employ the DP approach to DOE n both an optmal bacward recurson algorthm and a more tractable, forward, lmted loo-ahead algorthm that yelds near-optmal desgns. The latter s obtaned by mplementng a verson of the coordnate exchange algorthm (CEA) suggested by Feodorov (197). In order to reduce the ntal model-bas, we propose an upgrade procedure of the model order based on nformaton accumulatng through expermentaton. Our procedure elmnates the problematc assumpton (Box et al., 1978) that the mathematcal model that descrbes the physcal phenomenon s nown exactly a pror. Both algorthms are llustrated n detaled examples. A further contrbuton s the development of upper and lower bounds on the value of ncremental nformaton under certan assumptons. These bounds are ndependent of future experment outcomes and assst the desgner to estmate the mnmum and the maxmum beneft obtanable by an addtonal experment, and thus, to decde whether to contnue expermentng. The rest of the paper s organzed as follows. Secton defnes basc nformaton theory concepts, manly entropy and nformaton, and dscusses the evoluton of nformaton n sequental experments. Secton 3 ncludes a quanttatve descrpton of the evoluton of nformaton wth ncremental expermentaton and motvates a dynamc programmng (DP) approach to sequental DOE. Secton 4 provdes a general stochastc DP framewor for DOE. A detaled one-dmensonal analytc algorthm s developed n Secton 5 for near-optmal sequental DOE. The algorthm s based on a lmted loo-ahead approxmaton of the optmal DP soluton. The applcablty of sequental DOE to real problems s demonstrated n a numercal mplementaton of the algorthm n Secton 6 for a mult-dmensonal problem where the structure of the response model s not nown a pror and s selected by the algorthm. Secton 7 concludes the paper. K, denoted by IðY ; KÞ, ntroduced by Shannon (1948a,b) s defned as IY; ð KÞ ¼ HðKÞ HðKjY Þ ¼ Z fy;g fðy; Þlog f Y fðy; Þ ðyþf K ðþ dyd; where f Y ðyþ, f K ðþand fðy; Þ are the margnal and the jont probablty densty functons (pdf) of Y and K, respectvely, HðKÞ s the dfferental entropy of K defned as Z HðKÞ ¼ f K ðþlog f K ðþd; ð3þ fg and HðKjY Þ s the condtonal dfferental entropy of K gven Y whch s the expected value of the entropy of the condtonal dstrbuton, averaged over the condtonng random varable,.e., Z HðKjY Þ ¼ f ð; yþlogf K ðjyþdyd: ð4þ fy;g Thus, Shannon nterpreted nformaton as the reducton of the entropy of one r.v. condtoned by another r.v. and used the entropy as a measure of uncertanty, as llustrated for a Bernoull r.v. n Fg. 1. The fgure consders a dscrete r.v. tang one value wth probablty a and another value wth probablty 1 a. The bnary entropy functon, gven by hðþ¼ a a log a ð1 aþlogð1 aþ,s measured n bts (or shannons) f the log base s two. Followng the above defntons, a reasonable formulaton of the DOE tas s: determne that subset of experments, out of all possble combnatons of factor levels, whch maxmzes nformaton, namely solve max ½IY; ð KÞŠ; X where X belongs to a feasble desgn sets (condtons and constrants on feasblty of desgns can be found, for example, n Atnson and Donev, 199). In words, the DOE tas can be thought of as amng to maxmze, over ðþ. Evoluton of nformaton n sequental experments Consder a system descrbed by the model gven n Equaton (1). Let Y be a contnuous random varable (r.v.) representng the experment response and K be the random varable representng the estmator of an unnown characterstc of the system (henceforth, random varables, other than e, are denoted by captal, and ther realzatons by lower case). The nformaton n Y about Fg. 1. The bnary entropy of a dscrete random varable.
1090 Ben-Gal and Caramans a set of feasble desgns, the nformaton n the experment outcomes about an estmator of system characterstcs. Later t s shown that for lnear Gaussan models, the nformaton crteron produces desgns that are dentcal to the ones produced by tradtonal DOE alphabetc optmalty crtera (Atnson and Donev, 199). In a sequence of experments, one can consder the condtonal nformaton, gven by IY ð ; Kj Y 1 ;...; Y 1 Þ ¼ HðKjY 1 ;...; Y 1 Þ HðKjY 1 ;...; Y 1 ; Y Þ; ð5þ whch, n the context of expermental desgn, we nterpret as the ncremental nformaton ganed from the th experment response Y, gven the responses of prevous experments Y 1 ;...; Y 1. Snce nformaton satsfes the chan rule, the total nformaton, whch s ganed by a set of experments, can be expressed as follows: IY ð 1 ;...; Y K 1 ; Y K ; KÞ ¼ XK ¼1 IY ð ; KjY 1 ;...; Y ; Y 1 Þ: ð6þ The models consdered here are those descrbed by Equaton (1) and nclude addtve Gaussan nose terms. Hence, responses are normally dstrbuted. The Gaussan dstrbuton maxmzes the entropy over all dstrbutons wth the same covarance matrx (Cover and Thomas, 1991). Hence, the normal dstrbuton provdes us wth an upper bound on the uncertanty of a r.v. wth an unnown pdf. Moreover, the normal dstrbuton s wdely used n DOE and regresson models, and s practcally justfed n many stuatons by the central lmt theorem. It can be shown that maxmzaton of the nformaton measure n desgns that nclude nteractons requres applyng a desgn matrx wth a certan resoluton, n partcular, a resoluton ensurng that model terms are not alased wth each another. The problem of obtanng a desgn wth the hghest possble resoluton for a lmted number of experments s further addressed n Ben-Gal and Levtn (1998, 001) by applyng crtera that are based on the mathematcal correspondence between Error-Correctng Codes (ECC) and Fractonal Factoral Experments (FFE). 3. Decreasng returns of ncremental nformaton gatherng Consder the case where the expermenter wants to maxmze the nformaton about the model parameters (.e., K, n ths case, are the model parameters) through sequental expermentaton. Let Y be the response vector of the th experment n a multple lnear regresson model: Y ¼ X b þ e; ð7þ where X s the m p desgn matrx used n the th experment; e s a m-dmensonal vector of d Gaussan random varables wth zero mean and varance r ; and b s a p-dmensonal vector of unnown parameters. Let B be the maxmum lelhood estmator of b after the th experment, whch s also the least-squares estmator for the Gaussan case. It s well-nown (Myers and Montgomery, 1995) that B s p-varate normally dstrbuted:! 1!! 3 B ¼ X X 0 X X X 0 y X 1 N p 4b; X 0 X r 5; ¼1 ¼1 ð8þ where y s the response of the th experment. The a pror condtonal dstrbuton of Y can be estmated at tme 1 through Bayesan nference usng the prevous 1 responses. Y s an m-varate normal random varable Y jðy 1 ¼ y 1 ;...; Y 1 ¼ y 1 Þ! X 1 1! 3 N m X B 1 ; X 0 X X 4 X 0 X r 5: ¼1 Gven the jont pdf of the estmators and the responses, the ncremental nformaton and the total nformaton n the responses about the parameters are defned as follows (proofs are gven n Ben-Gal and Caramans (1999) and Ben-Gal and Levtn (001). Defnton 1. The ncremental nformaton n the responses about the parameters n a Gaussan lnear regresson model s gven by! 3 IðY ; BY j 1 ;...; Y 1 Þ ¼ 1 log det I_ p þ X 0 X X 1 1 4 X 0 X 5; ð9þ where _ I p s the p-dmensonal dentty matrx and det stands for the determnant of the matrx. Defnton. The total nformaton ganed from K expermental responses about the parameters n a Gaussan lnear regresson model s gven by ¼1 ¼1 IðY ;...; Y K 1 ; Y K ; BY j 1 Þ ¼ 1 log det X K X 0 X ¼1 ¼1 "! # X 0 1 X 1 1 : ð10þ Condtonng over Y 1 s done through Bayesan nference when the desgner has no advance nowledge about the pdf of B. The vector of the frst expermental response Y 1 enables the desgner to establsh a pror dstrbuton of B and proceed to update t by successve expermental responses. Observatons made n connecton to the above results assst n specfyng desgns that maxmze ncremental (or
Sequental DOE va dynamc programmng 1091 total) nformaton. We call nformaton-maxmzng desgns H-optmal desgns. Ben-Gal and Levtn (001) show that for a lnear regresson model wth an addtve Gaussan nose, the H-optmalty crteron and the wellnown D-optmalty crteron (Atnson and Donev, 199) concde, snce both tend to mnmze the scaled determnant of the varance matrx of B. Ths concdence should not be surprsng n the Gaussan case. It s smply due to the relaton between the dfferental entropy and the varance matrx. D-optmal desgns have been extensvely nvestgated n DOE lterature (Keefer and Wolfowtz, 1959; St. John and Draper, 1975; Hardn and Sloane, 1993). In partcular, t s nown that for multple lnear regresson models wth coded factors (.e., factors wth level range from 1 to 1), the D-optmalty crteron (and hence, n the Gaussan case, the H-optmalty crteron) requres the normalzed desgn matrx to be orthogonal so that all off-dagonal elements of X 0 X are zeros and the dagonal elements of X 0 X are as large as possble (Box and Draper, 1971; Myers and Montgomery, 1995). A second observaton consders the rate and cost of acqured nformaton n a sequental DOE approach and motvates the use of a dynamc-programmng-based desgn framewor. By applyng orthogonal desgns that satsfy X 0 X ¼ m _ I p to multple lnear regresson models wth coded factors, one obtans the followng ncremental nformaton out of Equaton (9): IðY ; BY j 1 ;...; Y 1 Þ ¼ 1 log det m I_ p ð 1Þm _ 1 I p ¼ p log ; ð11þ 1 and by Equaton (10), the total nformaton s IðY ;...; Y K 1 ; Y K ; BY j 1 Þ ¼ p log K; ð1þ whch s the maxmum amount of nformaton obtaned from a seres of K experments about the lnear regresson model. Note that the nformaton per estmator component s equal to 1= log K and ncreases at a slow logarthmc rate wth the number of experments (or equvalently, the ncremental nformaton decreases nversely proportonal to ). That s, as K!1 one can obtan an nfnte amount of nformaton (snce B s a contnuous r.v.), albet, at a decreasng rate. Ths mples that at a certan pont the cost of addtonal expermentaton wll outwegh the value of addtonal nformaton. The answer to the nterestng queston when should one stop expermentng? s complcated further when the value of r s unnown. Optmal stoppng rules for sequental DOE can be obtaned n prncple from the stochastc dynamc programmng problem formulaton presented next. 4. A stochastc dynamc programmng (DP) framewor In ths secton we consder a sequental desgn of experments approach. We develop optmal sequental DOE strateges by applyng the nformaton measure presented above. 4.1. Modfcaton of the standard DP problem notaton conventon We start by modfyng the standard DP problem notaton conventon to accommodate standard DOE notaton conventons. The matrx of control factor settngs (.e., the control varable) n the experments performed at tme s denoted by X ; the observatons at tme correspond to system responses or expermental outcomes and are denoted by y ; the observaton dsturbance at tme s denoted by e (e s generally consdered to be a constantvarance Gaussan random varable and the subscrpt s omtted); the system state s the system unnown parameters and s denoted by b ; the nose component s denoted by the standard symbol w, but w ¼ 0 for standard DOE models wth fxed parameters: b ¼ b. The suffcent statstc at tme s denoted by P. P can be ether the probablty dstrbuton of B, the estmator of b, or the probablty of an event,.e., a measure of an approprately defned subset of the range space of the estmator B. Fnally, the tme reference s modfed so that the control selected at tme, namely the desgned experment matrx X, affects the responses y assocated wth the end of the same tme perod nstead of the begnnng of perod þ 1. 4.. General formulaton of the DP approach to DOE We consder the stochastc dynamc programmng paradgm under mperfect state nformaton (Bertseas, 1995). Fgure presents the tme sequence of decsons and nformaton gatherng. The desgner has mperfect nowledge of the unobserved response surface parameters b. Thus, for system response functons wth fxed parame- Fg.. Decson and nformaton evoluton n a stochastc DP framewor for desgn of experments.
109 Ben-Gal and Caramans ters, b s consdered as a tme nvarant mperfectly observed state where b þ1 ¼ b ¼ b. Another nterestng approach s to consder a tme-dependent system state dentfcaton, where one starts wth a low-order model, and hence a low-dmenson b vector, and gradually upgrades ts order f the statstcal sgnfcance of hgher order terms s supported by the nformaton obtaned from past experments. For example, the ntal model may be a frst-order polynomal wth nsgnfcant quadratc terms that may become sgnfcant and support an ncrease of the model order n later stages. We llustrate such an approach n Secton 6 by mplementng the DP algorthm to a numerc mult-dmensonal example. In general, we note that nformaton s gathered by observng the system responses y through experments. K s an estmator of some characterstcs of the system, whose probablty dstrbuton depends on b, and whch s defned to represent our partal nowledge of the system characterstcs of nterest at tme. The system responses are determned by the generally tme-varyng functonal relaton: y ¼ f ðx ; e ; bþ; ð13þ where y s a vector of system responses at tme. X s the controlled desgn matrx at tme that belong to the set of feasble and allowable desgns, X N. e s the observaton dsturbance at tme. It s characterzed by a gven probablty dstrbuton P e ðjx ; b Þ, whch depends on the system parameters and the current controls and dsturbances. W s the nose random varable characterzed by the probablty dstrbuton P W ðjx ; b Þ, whch may depend explctly on X and b but not on pror realzatons of the nose and dsturbance varables: w 1 ;...; w 0, e 1 ;...; e 0. W s consdered only for systems wth tmevaryng parameters,.e., cases where system response parameters evolve over tme, such as when there s parameter drftng. Note that DOE tradtonally deals wth s a statstc of the system at. Note that when experment X þ1 s contemplated, y þ1 has not yet been observed and s a random varable whose probablty densty depends on past responses and current control, Pr y þ1 ji ; X þ1 ¼ Pr yþ1 jx 1 ; X ;...; X þ1 ; y 1 ;...; y : ð16þ Usually one can defne a suffcent statstc P ði Þ that represents all the relevant nformaton n I about those characterstcs of the system one s nterested n (e.g., the probablty of achevng a system response wthn a specfc requred tolerance). In partcular, the suffcent statstc s gven by the condtonal probablty dstrbuton P P K ji of an approprately defned r.v. K, whch taes values, and depends on the system parameters as estmated by past expermental responses. For example, P can represent the condtonal probablty dstrbuton of a bnary r.v. and, hence, the probablty of an event, or the condtonal probablty dstrbuton of B, the estmator of b, gven the nformaton matrx at tme,.e., P P B ji ðbþ. Ths latter example s used n the rest of ths paper. As addtonal nformaton s obtaned through experments, P s re-evaluated and updated recursvely, through a flter of the form: P ¼ / 1 ðp 1 ; y ; X Þ ¼ ^/ 1 ðy 1 ;...; y ; X 1 ;...; X Þ: ð17þ When, the suffcent statstc can be characterzed by a set of numbers whose cardnalty s tme nvarant, and therefore smaller, for all, than the monotoncally ncreasng cardnalty of the nformaton matrx I,ts easer to mplement a polcy that maps the suffcent statstcs to the acton space (Bertseas, 1995). Assume that the cost per stage can be expressed as a functon of the control X, and the suffcent statstc P. Thus, g ðp ; X þ1 Þ ¼ ( E g C X ð þ1; w þ1 ; Þ when X þ1 s the þ 1 desgn matrx, w þ1 ; ð Þ when X þ1 T ðtermnateþ. g T P ð18þ a system of fxed parameters, namely w ¼ 0 for all. Denote the nformaton avalable to the controller at tme by I and call t the nformaton matrx (Bertseas, 1995). I ¼ ðy ;...; y 1 ; X ;...; X 1 Þ; ð14þ thus, I ¼ ði 1 ; y ; X Þ; I 1 ¼ ðy 1 ; X 1 Þ ¼ 1;...; K: ð15þ We vew these equatons as descrbng the evoluton of the expermenter s nowledge about the system, where I The functon g C represents the cost of runnng an addtonal experment whle the functon g T represents the cost assocated wth uncertanty when expermentaton s termnated n a manner smlar to that proposed by Krehbel and Anderson (199). Note that estmaton of the expectaton over requres P. X þ1 s a matrx of real numbers f another set of experments s conducted. Otherwse, X þ1 s a logcal varable (flag). X þ1 T denotes termnaton of expermentaton. Usng Bellman s prncple of optmalty, the mperfect state nformaton DP can now be wrtten as:
Sequental DOE va dynamc programmng 1093 8 >< J ðp Þ ¼ mn mn X þ1 >: ð Þ g T P 9 ð ÞþJ þ1 / X þ1 ; y þ1 ; P ; >= >; E g C X þ1; w þ1 ; w þ1 ;y þ1 ; ¼ 1; ;...; K and X ¼ T at the fnal stage: ð19þ An optmal polcy, g 0 ; g 1 ;...; g K 1, mnmzes Equaton (19) by determnng the control at tme, X ¼ g 1ð P 1Þ, where g s an approprate functon mappng the suffcent statstc to the allowable control set, g ðp Þ N. The optmal polcy, mared by a star, can be obtaned recursvely by startng from the boundary condton J K ðp K Þ ¼ g T Kð P KÞ and then usng Equaton (17) to mnmze the rght-hand sde of Equaton (19) for every possble P K 1 to obtan g K 1ð P K 1Þ and contnue wth the DP bacward recurson untl J 1 ðp 1 Þs computed. The optmal cost J s then obtaned by calculatng J ¼ E J 1 / 0 g y1 0ð P 0Þ; y 1 ; P 0 : A specal but mportant case arses when the control,.e., the desgn of experments, must conform to a desgn resoluton constrant and a constrant n the mnmum number of levels that factors must be set at. In ths case, the optmal expermental desgn tas, whch arses at each recurson of the DP algorthm (.e., mn over X þ1 n Equaton (19)) can beneft from the results n Ben-Gal and Levtn (001). 4.3. The cost per stage functon and ts relaton to nformaton The cost per stage functon can be modeled to represent the costs and revenues assocated wth nformaton. For example, consder a system subject to Gaussan nose, where w ¼ 0, P ¼ C the varance matrx of B at tme, and g T ð P Þ c logðdet C Þ,.e., a logarthmc functon of the determnant multpled by a cost constant c. Ths s a reasonable representaton of cost snce t suggests that termnaton costs are proportonal to uncertanty. It follows from Equaton (19) that for a selected value of X þ1 an addtonal experment at tme s desrable when max X þ1 E g T yþ1 ð P Þ g T þ1 / X þ1 ; y þ1 ; P E g C X ð þ1; Þ > 0; ð0þ where the second term of Equaton (0) s the experment expected cost and the frst term of Equaton (0) s nothng but the revenue generated by the expected ncremental nformaton, snce under the above assumptons t follows (see Equaton (9)) that E g T y ð P Þ g T þ1 / X þ1 ; y þ1 ; P þ1 ¼ ce½logðdet C Þ logðdet C þ1 ÞŠ yþ1 ¼ c 0 E ½HðKjY 1 ;...; Y 1 ; Y Þ HðKjY 1 ;...; Y ; Y þ1 ÞŠ yþ1 ¼ c 0 IðY þ1 ; KjY 1 ;...; Y 1 ; Y Þ: ð1þ Thus, conductng an addtonal experment s desrable only f an experment can be desgned whose revenue from the expected ncremental nformaton exceeds the expected cost of the experment. Note that f the ncremental nformaton decreases monotoncally wth the number of experments, as n the case above where K B (n Equaton (11)), and f the cost of experments s tme ndependent, then, t s reasonable to apply a lmted loo-ahead algorthm. In fact, because of the monotoncally decreasng ncremental nformaton, a one-step loo-ahead algorthm corresponds to the open-loop feedbac control method (Bertseas, 1995). Ths leads to re-optmzaton of future decsons at each tme perod after replacng all future costs and benefts wth ther expected values (calculated on the bass of avalable nformaton), and assumng that no addtonal nformaton wll be made avalable n the future. The ratonale of the loo-ahead algorthm s that f the cost of an addtonal experment at tme s larger than the expected ncremental nformaton, then, on the average, the same wll be true at tme þ 1. The algorthm s llustrated further by an analytcal one-dmensonal example n Secton 5, and a numercal mult-dmensonal example n Secton 6. The nformaton-based defnton of the cost-to-go functon provdes addtonal justfcaton for usng the DP algorthm. Note from Equaton (8) that the varance of B s gven by C ¼ r ðx 0 XÞ 1, allowng standard DP results for quadratc cost functons to be appled. Moreover, the logarthmc functon s a monotonc ncreasng functon, whch mmcs the ncremental revenue ganed by an addtonal experment. Ths transformaton s consstent wth respect to the optmzaton problem, as argmn½logðþ x Š ¼ argmn½š, x and t has some appealng statstcal features as shown by Box (1988). Fnally, note that logarthmc costs are approprate n many stuatons for physcal reasons. The storage cost of a memory devce wth a q-ary alphabet s a case n pont; f U denotes the largest attanable value of the data, then, at most log q due encodng bts are needed.
1094 Ben-Gal and Caramans The ablty to assgn a monetary cost to uncertanty that s consstent wth the monetary cost of runnng experments s clearly a prerequste to the practcal value of the DOE framewor proposed n ths artcle. We brefly comment on several applcaton areas where costng uncertanty s standard practce. Materal handlng systems n wafer fabrcaton clean rooms consst of several desgn parameters, the most mportant of whch s the sze of the stocers to be placed n varous locatons of the clean room. Lengthy Monte Carlo smulatons of the clean room operaton are run assumng nfnte capacty stocers, and the mean stocer level and ts varance, or alternatvely, the maxmum stocer level and ts varance are estmated. The stocer s then szed accordng to rules of the type: stocer sze = average stocer level + (standard devaton of average stocer level), or stocer sze = max stocer level + *(standard devaton of max stocer level). The cost of a unt of standard devaton s then equal to the cost of a unt of stocer capacty dvded by or. Supply chans of nteractng supplers and buyers owe ther proftablty to the extent to whch they can operate wth low nventory levels or safety stocs. Experence and theory both pont towards the fact that the raw materal nventory n front of a supply chan ln s almost lnearly related to the sum of the coeffcents of varaton of the nter-arrval and processng tmes at that ln (Slver et al., 1998). Assocatng the coeffcents of varaton wth nformaton extracted from experments, and notng that the cost of holdng nventory due to obsolescence, degradaton and the opportunty cost of worng captal, s fnancally quantfed, the value of nformaton n monetary terms appears clear and wthn reach. Inventory management s another applcaton area where the economc value of nformaton s mportant. The lterature reports varous closed-form cost functons that descrbe the economc value of nformaton. For example, Slver et al. (1998) provde penalty functons for dfferent nventory control rules that depend on the standard devaton of the demand lead-tme and report on numercal llustratons of the penalty cost where the standard devaton s one of the parameters. Fnally, Krehbel and Anderson (199) use a penalty loss functon, whch s assocated wth the varance of unnown parameters n the context of engneerng product and process desgn. They propose the use of a quadratc loss functon of the type ntroduced by Taguch (1978) and Taguch and Clausng (1990). 5. Analytc algorthm: a lmted loo-ahead stoppng rule for a one-dmensonal model Consder the smple one-dmensonal model wth a fxed nose component, y ¼ x b þ e; where e N 0; r : ðþ We start ths example by assumng that r s nown. Ths assumpton s relaxed later. We defne r.v. K BjI,.e., mplyngp P B ji and w ¼ 0. We use a compact notaton: ¼1 x P x and P ¼1 x y P x y. Note that at tme these expressons are nown constants. The maxmum lelhood estmator of b at tme s normally dstrbuted,! B ¼ X. X x y x N b; r. X x : ð3þ Followng Equatons (17) and (3), the mean and the varance of B are defned as suffcent statstcs. The probablty dstrbuton of B s updated after observng the ð þ 1Þth response as follows: P x y þ x þ1 y þ1 EB ½ þ1 Š ¼ / l ðb ; x þ1 ; y þ1 Þ ¼ P ; x þ x þ1 r VB ½ þ1 Š ¼ / S ðb ; x þ1 ; rþ ¼ P : ð4þ x þ x þ1 Recall from Equaton (9) that, for a gven r, the future value of the ncremental nformaton can be pre-calculated as a functon of future control factors solely. Thus, at tme, the ncremental nformaton s a functon of x þ1 only, IY ð þ1 ; B Þ ¼ 1 log ð VB ½ Š=V½B þ1 ŠÞ ¼ 1 log 1 þ x þ1 = P x : ð5þ Let the termnaton cost at the th step be g T ð VB ½ ŠÞ ¼ c logðvb ½ ŠÞ, where c s a fxed uncertanty cost rate expressed n unts of dollars per bts. The experment cost at tme s assumed to be lnear n the magntude of the control, namely g C ð x þ1þ ¼ c 1 jx þ1 j, where c 1 s the experment cost per unt of the control magntude. Such a cost structure mples an experment, where hgher values of the control factors are consdered to be more expensve. Consder a fnte horzon bacward DP algorthm, where ¼ 1; ;...; K. The termnaton cost at tme K s J K ðvb ½ K ŠÞ ¼ g T Kð VB ½ KŠÞ ¼ c logðvb ½ K ŠÞ, and the cost-togo functon at the th step s J ðvb ½ ŠÞ ¼ mn mn½c 1 jx þ1 jþ J þ1 ð/ ðvb ½ Š; x þ1 ÞÞŠ; c logðvb ½ ŠÞ : x þ1 ð6þ Thus, by nvestng c 1 jx þ1 j at the th step, the desgner decreases the termnaton cost at least by the value of ncremental nformaton obtanable from an addtonal experment, c ðlogðvb ½ ŠÞ logðvb ½ þ1 ŠÞÞ ¼ c IY ð þ1 ; BY j 1 ;...; Y 1 ; Y Þ: ð7þ
Sequental DOE va dynamc programmng 1095 The frst mnmzaton operator n Equaton (6) optmzes the number of experments wth respect to experment costs and revenues of ncremental nformaton, whch decreases wth tme. The second mnmzaton operator (nsde the curly bracets) optmzes the value of the control varable f an addtonal experment s to be conducted. We now suggest a conservatve near-optmal one-step loo-ahead heurstc for the on-lne forward DP algorthm. Note agan that the expected ncremental nformaton decreases monotoncally wth the number of experments (Equaton (11)), therefore, f the cost of an addtonal experment s larger than the expected ncremental nformaton revenue n the th experment, then, t s larger than the expected ncremental nformaton n the ð þ 1Þth experment. Accordngly, at each step, the desgner consders whether to contnue expermentng by comparng the cost of an addtonal experment wth the mnmum expected revenue generated by the ncremental nformaton from the next experment. That s, the cost functon at step s wrtten smply as J ðvb ½ ŠÞ mn mn½c 1 jx þ1 jþ c logðvb ½ þ1 ŠÞŠ; c logðvb ½ ŠÞ : x þ1 ð8þ If an addtonal experment s carred out, the value of the control factor s optmzed as llustrated n Fg. 3. Note that such an algorthm s practcally applcable, although n the absence of the strct monotoncty assumpton t may cause a premature stop of the experment, snce the beneft of the ncremental nformaton gathered from future experments s not taen nto account. Applyng such an algorthm wth a lmted loo-ahead of more than one experment can mprove ts accuracy. For llustratve purpose, we consder a further smplfcaton by assumng a fxed postve value of the control and a fxed number of experments. Then, the DP reduces to a closed-form, open-loop mnmzaton problem, where the total cost functon s gven by Fg. 3. A schematc llustraton of the cost functon of an addtonal experment at tme. The value of the control s optmzed to support a sngle step loo-ahead DP algorthm. J ðx; KÞ ¼ ðk 1Þx c 1 þ c log r =Kx ; ð9þ whch yelds an optmal soluton as a functon of K for a fxed value of x, and a lmted loo-ahead soluton as a functon of x for a fxed value of K: h ðkþ ¼ c þ log c 1 ðk 1Þ r =4c K ; where J x x ¼ c =c 1 ðk 1Þ; J ðxþ ¼c 1 þ logðc 1 r =c xþ c1 x; where K ¼ c =c 1 x; ð30þ as llustrated by Fg. 4(a and b) respectvely. Next, we present an on-lne DP algorthm, whch s applcable when r s unnown and has to be estmated and updated after each experment. Note that at tme the unbased maxmum lelhood estmator of r s gven by the sample varance. Applyng Equatons (3) and (4) the expermental nose and the varance of B are estmated as follows: P ^r ¼ 1 X x P ðy x BÞ y P x y ¼ 1 ¼1 ð 1Þ P ; x P x P y P x y VB ½ Š ¼ ð 1Þ P : ð31þ x Fg. 4. (a) The total cost functon J x ðkþ, for a fxed value of the control factor (x) as a functon of the number of experments (K), where c 1 ¼ 1 and c ¼ 4; and (b) the total cost functon J K ðþ, x for a fxed number of experments (K) as a functon of the control factor value (x), where c 1 ¼ 1 and c ¼.
1096 Ben-Gal and Caramans Smlarly to the prevous example, the varance of B decreases as the values of the control factor ncrease. Namely, multplyng each control factor by a postve constant c results n a reducton of the estmator varance by an order of 1=c. Note, however, that now the future values of the varance of B can not be pre-calculated, as was done n the prevous example, snce they explctly depend on future response values. Accordngly, one has to consder an on-lne DP algorthm based on the expected ncremental nformaton wth respect to future responses. Usng Equaton (1) the ncremental nformaton at tme, for ths case, s gven by " IY ð þ1 ; B Þ ¼ E yþ1 0 B @ 1 log VB ½ P P y þ yþ1 x þ x Š P x þ x þ1 þ1 13 C7 P A5; x y þ x þ1 y þ1 ð3þ whch s a functon of x þ1 wth mnmum at x þ1 ¼ 0 (where y þ1 ¼ e). The expectaton s taen wth respect to the future experment response y þ1. Note that at tme, Y þ1 s a Gaussan random varable whose pdf depends on x þ1, past responses and past controls as follows: Y þ1 ðx þ1 Þjðy 1 ;...; y ; x 1 ;...; x Þ P 3 x þ x þ1 ^r þ1 N4x þ1^b þ1 ; P 5: ð33þ x In partcular, by applyng Equatons (3) and (31) t follows that Y þ1 ðx þ1 Þjðy 1 ;...; y ; x 1 ;...; x Þ 0 P N x P P x þ x þ1 x P y P 1 x þ1 x y y B P ; x ð 1Þ P C @ A : x ð34þ Accordngly, the on-lne one-step loo-ahead DP algorthm for postve controls s: whch mples, once agan, that one conducts an addtonal experment, payng the cost c 1 x 1, only f the revenue assocated wth the expected ncremental nformaton s hgher than the experment costs, namely, f the followng condton s satsfed: c E ½I ðy þ1 ; B ÞŠ ¼ c log VB ½ Š yþ1 E ½c logðvb ½ þ1 ŠÞŠ c 1 x 1 ; ð36þ yþ1 where the star n I ðy þ1 ; B Þ denotes an optmal selecton of the control factor x þ1 that maxmzes the net revenue of an addtonal experment. Equaton (36) can be computed numercally for gven response values. Moreover, we suggest both an upper bound and a lower bound on the value of ncremental nformaton. These bounds are ndependent of future responses and may assst the desgner to evaluate the mnmum and the maxmum beneft obtanable by an addtonal experment, and thus determne whether to contnue expermentng. We derved these lower and upper bounds by applyng, respectvely, the Jensen s nequalty or utlzng a Taylor seres expanson about Ey ½ þ1 Š (proofs are gven n Ben-Gal and Caramans (1999)):! 0P 1 1 log 1 þ x þ1 P IY ð þ1 ; B Þ 1 x x log þ x þ1 @ P A x 1 1 þ ð 1Þ : ð37þ Note that the bounds approach each other, at a rate proportonal to 1=, whch s equvalent to the decrease rate of ncremental nformaton. A smlar approach can be used n the mult-dmensonal case when observaton error s present. 6. Numercal example: a lmted loo-ahead stoppng rule for a mult-dmensonal model In ths secton, we mplement the lmted loo-ahead algorthm to the mult-dmensonal model gven n Equaton (7), where the control s a desgn matrx. The J ðvb ½ ŠÞ mn mn E g C x ð x þ1þþg T þ1 VB ð ½ Š; y þ1 ; x þ1 Þ ; g T ð VB ½ ŠÞ þ1 yþ1 8 ( " P P x mn E c 1 x þ1 þ c log þx P þ1 y þy #) 9 þ1 x y þx þ1 y þ1 P ; >< x þ1 yþ1 ð 1Þ x þx >= þ1 ¼ mn P P x P c log y ; x y >:!; >; P ð 1Þ x ð35þ
Sequental DOE va dynamc programmng 1097 Fg. 5. The lmted loo-ahead forward algorthm for the mult-dmensonal case. forward lmted loo-ahead algorthm s presented n Fg. 5. It conssts of two parts both usng the Coordnate Exchange Algorthm (CEA) suggested orgnally by Feodorov (197). Orgnally, the CEA was desgned to generate D-optmal desgns by mprovng a startng desgn and by mang ncremental changes to ndvdual elements of the desgn matrx. At each teraton, the CEA consders both addton (as suggested by Box and Hunter (1965)) and deleton of desgn ponts. Among all possble exchanges of pars of ponts t selects the one that generates the greatest ncrease n the determnant of the nformaton matrx. The CEA constructs D-optmal desgns based on three nputs: () the number of control factors, n; () the model order (e.g., man-effects, lnear wth nteractons, and quadratc ); and () the requred number of experments, m. Snce under the Gaussan assumpton the H-optmalty and the D-optmalty crtera concde, the CEA s used to generate H-optmal desgns.
1098 Ben-Gal and Caramans In the ntal part, the CEA s appled to generate an ntal H-optmal desgn. The outputs of ths part are the ntal entropy, H 0, and the ntal desgn matrx, X 0 wth m 0 experments supportng the lower model order that s assumed at that pont. In the second part, the lmted loo-ahead algorthm uses the CEA to generate an economc-optmal desgn wth respect to the expected nformaton revenues and the expected expermental costs, as dscussed n Secton 4. and Equaton (19). The algorthm n ths part s performed teratvely (see Fg. 5). Gven a model order and the number of factors, the algorthm generates the smallest feasble desgn matrx, contanng m 1 experments (.e., enough experments to estmate all the model parameters). Then, usng a gven cost functon, the algorthm calculates the net expected revenue by consderng both the expected revenues of ncremental nformaton n unts of dollars per bt, as well as the expected expermental costs measured n unts of dollars per experment. At each step, a larger desgn, n whch m 1 s augmented by d new experments (m 1 :¼ m 1 þ d) s consdered. CEA s then employed to generate a new H-optmal desgn. The net expected revenue of the augmented desgn s calculated and compared wth the net expected revenue of the old desgn. If the new desgn s more proftable, the algorthm selects t as the best desgn thus far and the next teraton begns. If the new desgn s less proftable, the algorthm stops and declares the best desgn found thus far as the optmal one for the current model order. The suggested forward algorthm s ndependent of the cost functon and the step-sze (of course, a step-sze of d ¼ 1 s the most accurate step-sze, although t requres hgher computatonal effort). The suggested algorthm has another appealng feature. At each teraton, after obtanng the desgn matrx that maxmzes the expected net revenue for the current model order, the algorthm examnes a model upgrade. Provded that statstcal analyss supports the sgnfcance of the model upgrade, ts economc desrablty s evaluated. One can use a model upgrade procedure to ntroduce an adaptaton towards hgher order terms of the model. For example, one can select the ntal model as lnear, wth quadratc term coeffcents nsgnfcantly small. Then, as nformaton s gathered, f quadratc terms are observed to have a sgnfcant mpact on the response, one can nvestgate an ncrease of the model order. The model upgrade procedure reduces the ntal model-bas of the lmted looahead algorthm: nstead of assumng a nown model from the begnnng, the algorthm starts wth a low-order model ncreasng t gradually, n a fashon smlar to that of the RSM. When the order of the model ncreases, the amount of nformaton obtaned for a gven (fxed) number of experments wll ncrease snce more parameters can be now estmated. For ths reason, the algorthm consders a model upgrade at a pont when t s no longer economcal to conduct more experments under the current model. Durng the model upgrade process, the algorthm calculates the ncremental nformaton n each teraton by applyng an augmented verson of the CEA. The augmented CEA allows selecton of addtonal experments optmally, gven the augmented desgn matrx of the experments performed n the prevous stages (for more detals see n Atnson and Donev (199)). Table 1 presents the results of a numercal experment wth the lmted loo-ahead algorthm mplemented usng the MATLAB statstcal toolbox. For smplcty, we consder an ntutve cost functon: the net expected revenue s the dfference of expected ncome mnus excepted costs. We calculate the expected ncome by multplyng the expected ncremental nformaton by the constant ncome rate c 1 ¼ 10ð$=btsÞ, and compute the expected cost by multplyng the number of augmented experments, m 1, by cost rate c ¼ 1orc ¼ 1: ($/experment). The number of factors examned ranges from three to sx factors. The nteger step-sze used s d ¼ 1,.e., the experment sze at each teraton ncreases by a sngle run. We consder several models, ncludng man effects, man effects wth nteractons and quadratc. Table 1 presents the number of mnmum requred ntal runs m 0, the number of optmally augmented runs m 1, the ncremental nformaton and the total expected revenue n each case. The number of optmally augmented experments, m 1, and as a result the desgn matrx, X 1, are affected by the number of factors and are very senstve to the order of the model. The number of addtonal experments s (not surprsngly) senstve to the cost ratos. However, t was reassurng to observe that the ntal number of experments has almost no effect on the total number of experments performed by the proposed algorthm. Fgure 6 presents a senstvty analyss of expected revenues as a functon of the expermental sze for three factors and dfferent order models. It also presents the process of upgradng the model order by usng optmal augmentaton of experments. Revenues are plotted aganst the number of experments for three dfferent models separately: man effects, man effects wth nteractons and quadratc for the same cost functon consdered above wth cost rates c 1 ¼ 10 ($/bt) and c ¼ 1: ($/experment). Note that the ncremental nformaton for a gven number of experments ncreases wth the model order, resultng n hgher expected net revenue. Thus, the optmal number of experments grows wth the model order. The model upgrade lne represents the process of gradually upgradng the model order, usng the augmented CEA. Partcularly, n ths example, we upgrade the model order once the optmal desgn for the current model order s found and t s no longer proftable to ncrease the sze of the set of experments n current
Sequental DOE va dynamc programmng 1099 Table 1. Numercal results on ncremental nformaton and expected revenues for dfferent numbers of factors, models and cost rates Model type and cost constrants Number of factors n Intal number of mnmum runs requred m 0 Optmal augmented experment sze m 1 Inc. nformaton Net revenue ($) Man effects 3 5 19 4.67 3.89 c 1 ¼ 10; c ¼ 1: 4 6 4 6.09 3.13 5 7 9 7.48 40.09 6 8 34 8.69 46.14 Interactons 3 7 37 10.49 56.04 c 1 ¼ 10; c ¼ 1: 4 11 39 16.51 87.19 5 16 80.08 14.77 6 50 3.13 171.3 Quadratc 3 10 50 14.80 78.96 c 1 ¼ 10; c ¼ 1: 4 15 71 0.67 13.97 5 1 95 7.3 159. 6 8 13 38.53 6.95 Man effects 3 5 5 5.3 8.04 c 1 ¼ 10; c ¼ 1:0 Interactons 3 7 45 10.90 64.0 c 1 =10; c =1.0 Quadratc 3 10 56 14.63 90.36 c 1 ¼ 10; c ¼ 1:0 7. Concluson Fg. 6. Expected net revenues for H-optmal desgns wth an augmented number of experments. Assumng n ¼ 3 factors; a model upgradng procedure for dfferent model orders: man effects, nteractons, quadratc ; and cost constrants c 1 ¼ 10, c ¼ 1:. DOE. We assume that two model-order upgrades are performed followng statstcal evdence that addtonal coeffcents are sgnfcant. Note that almost no revenues are lost durng ths upgradng process, snce the augmented CEA accumulates nformaton effcently (that s, wth few teratons the model upgrade lne s very close to the hgher order model lne). Ths s the reason, for example, that the maxmum revenue of $78.96 s obtaned wth a fxed quadratc model usng a desgn matrx wth 50 experments, whle a revenue of $78.1 s obtaned for the same number of experments by upgradng from the nteracton model to the quadratc model, and usng the augmented CEA to add teratvely 15 experments to the best desgn obtaned for the nteracton model. We proposed a dynamc programmng framewor for sequental desgn of experments. The DP algorthm optmzes both the number of experments and the actual desgn of each experment. The number of experments s optmzed snce t s shown that the ncremental nformaton, whch s ganed from addtonal experments, s decreasng wth the number of experments (as 1= for orthogonal coded desgns). The value of the control (.e., the desgn of the next experment) s optmzed at each step of the algorthm wth respect to the trade-off between the cost of an addtonal experment and the revenue generated by ts ncremental nformaton. We suggested a stochastc DP approach to DOE and appled t by both a bacward algorthm and a forward, lmted loo-ahead algorthm, whch s practcally convenent and on the average near-optmal. We used both an analytc one-dmensonal mplementaton and a numercal mult-dmensonal mplementaton to llustrate the proposed approach. References Atnson, A.C. and Donev, A.N. (199) Optmum Expermental Desgn, Clarendon Press, Oxford, UK. Ben-Gal, I., Braha, D. and Mamon, O. (1999) A probablstc sequental methodology for desgnng a factoral system wth mul-
1100 Ben-Gal and Caramans tple responses. Internatonal Journal of Producton Research, 37(1), 703 74. Ben-Gal, I. and Caramans, M. (1999) A stochastc dynamc programmng framewor for an sequental desgn of experments full verson. Unpublshed Mmeograph seres No. 3B-98, CIM Lab, Tel Avv Unversty. Ben-Gal, I. and Levtn, L. (1998) Bounds on code dstance and effcent fractonal factoral experments, n Proceedngs of the IEEE Internatonal Symposum on Informaton Theory, IEEE Inc., Pscataway, NJ, USA. p. 436. Ben-Gal, I. and Levtn, L. (001) Applcaton of nformaton theory and error-correctng codes to fractonal factoral experments. Journal of Statstcal Plannng and Inference, 9(11), 67 8. Berlner, L.M. (1987) Bayesan control n mxture models. Technometrcs, 9(4), 455 460. Bernardo, J.M. (1979) Expected nformaton as expected utlty. Annals of Statstcs, 7(3), 686 690. Bertseas, D.P. (1995) Dynamc Programmng and Optmal Control, Athena Scentfc, Boston, MA. Box, G.E. (1988) Sgnal-to-nose ratos, performance crtera and transformatons. Technometrcs, 30(1), 1 40. Box, G.E. (199) Sequental expermentaton and sequental assembly of desgns. Qualty Engneerng, 5(), 31 330. Box, G.E. and Draper, N.R. (1971) Fractonal desgns, the jx 0 X j crteron, and some related matters. Technometrcs, 13(4), 731 74. Box, G.E. and Draper, N.R. (1987) Emprcal Model-Buldng and Response Surfaces, Wley, New Yor, NY. Box, G.E. and Hunter, W.G. (1965) Sequental desgn of experments for nonlnear models, n Proceedngs of the IBM Scentfc Computng Symposum n Statstcs, pp. 113 137. Box, G.E.P., Hunter, W.G. and Hunter, J.S. (1978) Statstcs for Expermenters, Wley, New Yor, NY. Bradt, R.N. and Karln, S. (1956) On the desgn and comparson of certan dchotomous experments. Annals of Mathematcal Statstcs, 7, 390 409. Cover, T.M. and Thomas, J.A. (1991) Elements of Informaton Theory, Wley, New Yor, NY. DeGroot, M.H. (196) Uncertanty, nformaton and sequental experments. Annals of Mathematcal Statstcs, 33(), 404 419. Feodorov, V.V. (197) Theory of Optmal Experments, Academc Press, New Yor, NY. Hardn, R.H. and Sloane, N.J.A. (1993) A new approach to the constructon of optmal desgns. Journal of Statstcal Plannng and Inference, 37, 339 369. Hardwc, J.P. and Stout, Q.F. (1995) Determnng optmal few-stage allocaton procedures, n Proceedngs of the 7th Symposum on the Interface Statstcs and Manufacturng wth Subthemes n Envronmental Statstcs Graphcs and Imagng, Interface Found, VA, USA, 7, pp. 34 346. Keefer, J. and Wolfowtz, J. (1959) Optmum desgns n regresson problems. Annals of Mathematcal Statstcs, 30, 71 94. Krehbel, T.C. and Anderson, D.A. (199) The use of a monetary loss functon to determne the optmal fractonal replcate of fractonal experments. Communcatons n Statstcs Theory and Methods, 1(8), 383 398. Lndley, D.V. (1956) On a measure provded by an experment. Annals of Mathematcal Statstcs, 7, 986 1005. Lucas, J.M. (1976) Whch response surfaces desgn s best. Technometrcs, 18, 411 417. Myers, H.R. and Montgomery, D.C. (1995) Response Surface Methodology, Wley, New Yor, NY. Shannon, C.E. (1948a,b) The mathematcal theory of communcaton. Bell System Techncal Journal, 7, 379 43, 63 656. Slver, E.A., Pye, D.F. and Peterson, R. (1998) Inventory Management and Producton Plannng and Schedulng, Wley, New Yor, NY. St. John, R.C. and Draper, N.R. (1975) D-optmalty for regresson desgns: a revew. Technometrcs, 17, 15 3. Taguch, G. (1978) Off-lne and on-lne qualty control systems, n Proceedngs of the Internatonal Conference on Qualty Control, Toyo, Japan. Taguch, G. and Clausng, D. (1990) Robust qualty. Harvard Busness Revew, 90(1), 65 67. Bographes Irad Ben-Gal s an Assstant Professor at the Department of Industral Engneerng n Tel Avv Unversty. He holds a B.Sc. (199) degree from Tel Avv Unversty, M.Sc. (1996) and Ph.D. (1998) degrees from Boston Unversty. He s a member of the Insttute for Operatons Research and Management Scences (INFORMS) and s the head of the Computer Integrated Manufacturng (CIM) laboratory at Tel Avv Unversty. He has wored for several years n varous ndustral organzatons, both as a consultant and as a project manager. Hs research nterests nclude qualty control, desgn of experments, testng procedures, and applcaton of nformaton theory to ndustral and bonformatcs problems. Mchael C. Caramans holds a B.S. n Chemcal Engneerng from Stanford Unversty and an M.S. and Ph.D. n Engneerng and Decson and Control from Harvard Unversty. He s Professor of Manufacturng Engneerng, Assocate Department Char, and Drector of the Center for Informaton and Systems Engneerng (CISE) at Boston Unversty. He has been prncpal nvestgator and project drector of a number of research projects sponsored by the Natonal Scence Foundaton, and has collaborated wth ndustry. Hs current research nterests are n the area of producton plannng and control of complex manufacturng systems. Contrbuted by the Desgn of Experments and Robust Desgn Department