A HILL-CLIMBING COMBINATORIAL ALGORITHM FOR CONSTRUCTING N-POINT D-OPTIMAL EXACT DESIGNS

Transcription

1 J. Sa. Appl. Pro., o., Journal of Saisis Appliaions & Probabiliy An Inernaional SP aural Sienes Publishing Cor. A HILL-CLIMBIG COMBIATORIAL ALGORITHM FOR COSTRUCTIG -POIT D-OPTIMAL EXACT DESIGS Mary Iwundu and Polyarp Chigbu Deparmen of Mahemais and Saisis, Universiy of Por Harour, Por Harour, Rivers Sae, igeria Address:: mary_iwundu@yahoo.om Deparmen Of Saisis, Universiy Of igeria, sua, Enugu Sae, igeria Address:: pe8higbu@yahoo.om Reeived: Jan. 3,, Revised April 4, ; Aeped April 8, Absra: A mehod ha maes use of ombinaoris for seleing objes ou of disinguishable objes is developed for onsruing D-opimal -poin ea designs. The diffiulies whih are eperiened in he variane ehange algorihms for onsruing D-opimal ea designs, suh as yling, slow onvergene and failure o onverge o he desired opimum, are no eperiened by his mehod. The mehod onverges rapidly and absoluely o he desired -poin D-opimal design and is effeive for deermining opimal designs in blo eperimens as well as in non-blo eperimens for finie or infinie number of suppor poins in he spae of rials. Keywords: D-opimaliy, Ea design, Blo eperimens, on-blo eperimens. Inroduion Given he eperimenal spae, { X ~, F, }, he problem in his wor is o develop an algorihm for onsruing an -poin D-opimal ea design measure w = w w i = ( i, i,, ni ) X ~ ; w i = for all i where, X ~ is an n-dimensional spae of rials whih is ompa, oninuous and meri (see Onuogu; 997). The spae of rials, X ~, shall be onsidered as having a regular or an irregular geomeri area and shall onsis of suppor poins, {,,, }, where represens he number of suppor poins in a finie spae of rials. By regular geomeri area we imply a geomeri area ha has a single simple mahemaial formula for ompuing is area. By irregular geomeri area we imply a geomeri area ha does no have a single simple mahemaial formula for ompuing is area. By finie spae of rials we imply a spae of rials for whih he underlying poinse is finie. Tha is, a spae for whih here are only finiely many poins.

2 34 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG - F = { f() } is a linear spae of finie dimensional oninuous funion defined on X ~. is a spae of random observaion error defined on X ~. Following Kiefer and Wolfowiz (959), we assume ha a eah poin, X ~, a random variable, namely, y = f() + is defined and is suh ha E(y ) = f() where = (,,, p ) is a p veor of unnown parameers whih are esimaed on he basis of unorrelaed observaions; is he random addiive error assoiaed wih y and is independen and idenially disribued wih E() = and E( ) = ; a onsan. We also assume ha Var (y ) = (normalized for onveniene = ), Cov ( y, y ) = ;,, X ~ ( ). In defining eperimenal designs, i is imporan o disinguish beween ea designs and oninuous designs. Aording o Coo and ahsheim (98), a design is an -poin ea design if is a probabiliy measure on X ~ whih aahes a mass o eah poin of he design and is a non-negaive ineger for X ~. We shall denoe he spae of -poin ea designs on X ~ by oninuous design is a probabiliy measure on X ~ suh ha d. ~ X X ~. On he oher hand, a The measure is an elemen of he spae, X ~, of probabiliy measure on X ~ and need no be an ineger. The above saed problem is a ombinaorial one; a problem of hoosing ou of in X ~ suh ha he deerminan of he informaion mari of he design is maimized. The informaion mari, M( ), of he design is given by M( ) = XX where X is an p design mari of, whose i h row is f( i ). Aording o Karlin and Sudden (966), he deerminan value of he informaion mari is a simple measure of he magniude of he informaion () mari. Thus, if M ( ) and M ( ) are wo pp non-singular informaion maries assoiaed wih () and, respeively, () () M ( ) M ( ) M ( ) M ( ) (see Onuogu (997), pg. 69). The maimizaion of he deerminan of a real-valued non-singular pp informaion mari, say, m m m p M = m m `m p m p m p m pp

3 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG 35 is ahieved when m jj ; j =,,, p are maimized and m jj ; j j are minimized; Onuogu and Chigbu (). A design, M( ), is maimized over all, is said o be a D-opimal ea design if he deerminan of he informaion mari, X ~ (see Coo and ahsheim (98)). When he eperimen is o be performed in b inomplee blos of sizes,,, b respeively, we see an -poin D-opimal design b = b bb ij = ( ij, ij,, nij ) X ~ ; i =,,, j, j =,,, b, = b j j Defining X as an p oeffiien mari for reamens and X B as an b indiaor mari for blos wih and elemens, (p + b) design mari is X ( p b) X p. X b B and he informaion mari o be maimized is M( ) = XX XX ( X X ) ( X X ) B B B B. Lieraure Review The ehange algorihms ied above beome slow due o he need o follow eah. They have been of muh usefulness in he onsruion of D-opimal ea designs. One of he earlies of suh algorihms is due o Mihell-Miller (97). The algorihm begins wih a randomly hosen -run design and moves in he direion of inreasing value of he deerminan of informaion mari. The proedure involves wo sages a eah ieraion by firs adding an ( + ) s run o he iniial design and hen subraing from he resuling design he poin ha leads o he minimum possible derease in deerminan value of informaion mari. The algorihm sops when here is no furher improvemen in he deerminan value or when he same poin is deleed and hen re-enered. The poin added orresponds o poin of maimum variane of prediion over X ~ and he poin deleed orresponds o he poin of minimum variane of prediion. The algorihm of Van Shalwy (97) is similar o ha of Mihell-Miller (97) bu firs delees a eah ieraion he poin in he iniial -poin design wih minimum variane of prediion. The -poin design is hen reovered by adding a poin from X ~ ha gives a maimum inrease in he deerminan value of informaion mari. The algorihm sops when he same poin deleed is aferwards re-enered. An appliaion of he Van Shalwy (97) algorihm in omparison wih he Mihell-Miller (97) algorihm on he problem of onsruing a 7-poin D-opimal ea design over he spae, X ~ ; - X ~, for a bivariae quadrai surfae, f(, ) = a + a + a + a + a + a +, reveals ha he number of variane evaluaions are redued by no less han 5%. Alhough he Van Shalwy algorihm in appliaion is more dire. However, boh algorihms are ompuaionally demanding due o he need o updae he designs and evaluae he deerminan a eah ieraion.

4 36 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG - The ehange algorihm of Fedorov (97), pg. 64 begins wih an -poin design and a eah ieraion evaluaes all possible ehanges of he pairs of poins,, from he design and l from he se of andidae poins. The ehange giving he maimum inrease in he deerminan value of informaion mari is onsidered. The proedure oninues as long as an inerhange inreases he deerminan. The Fedorov s ehange algorihm is made slow by he large number of poins o be onsidered a eah ieraion. As a way of speeding up he Fedorov s algorihm, many modifiaions have been suggesed (see e.g. Coo and ahsheim (98), and Johnson and ahsheim (983), Ainson and Donev (99). Eah of hese modifiaions aims a reduing he number of poins o be onsidered for ehange. For eample he KL algorihm of Ainson and Donev (99), pg. 73 ehanges poins in he design having relaively low variane of prediion wih andidae poins for whih varianes are relaively high. The algorihm begins wih an -poin design and moves one sep a a ime in he direion of inreasing value of deerminan of informaion mari. When here is no longer any ehange ha would inrease he deerminan value, he algorihm erminaes. The KL algorihm is followed by an adjusmen algorihm whih searhes away from he andidae lis (see Ainson and Donev (99), pg ). They also invesigaed he properies of opimum designs when here are boh qualiaive faors (represened by he blos) and quaniaive faors. The modified algorihm (BLKL) provides he opporuniy o divide he eperimenal rials ino blos of speified sizes (see Alinson and Donev (989, 99)). The DETMAX algorihm of Mihell (974) hough slighly differen from he ehange algorihms desribed above is a generalized version of he Mihell-Miller algorihm. I begins wih a randomly hosen -run design and a pre-speified lis of andidae se. A hosen number of poins is sequenially added o and hen deleed from he design (following is eursion sheme) hus improving he saring design. In DETMAX, he requiremen ha an ( + )-poin design be reurned immediaely o an -poin design is relaed. The algorihm maes use of posiive and negaive eursions during whih he size of he design may vary from o + K and from o K, respeively, where K is a user-seleed ineger onsan. The DETMAX algorihm is run several imes ( imes) in eah ase, eah ime saring wih a differen randomly seleed iniial -run design and he Bes DETMAX design is ha whih gives he larges deerminan value in he number of ries. When an eursion size reahes K, he algorihm erminaes. The ehange algorihms ied above beome slow due o he need o follow eah suessful ehange by updaing he design, he informaion mari, he variane-ovariane mari, he varianes of he predied values a he design and andidae poins and he evaluaion of respeive deerminan values of informaion maries. Moreover, hese algorihms are based on he variane of he predied response and hene, have he high probabiliy of geing rapped a a loal opimum. We presen a new mehod of onsruing D-opimal designs. The mehod is based on he ombinaoris of he suppor poins ha mae up X ~ and is suh ha for given { X ~, F, }, he suppor poins ha mae up X ~ an be arranged ino onenri balls (groups) and an opimal ombinaion of hese balls an be obained so ha he orresponding -poin ea design measure is D-opimal. An araive feaure of his mehod is ha by grouping i is easy o idenify ses of informaion maries wih equal diagonal elemens, so as o ompare he informaion maries wihin a given se on he basis of he off-diagonal elemens wihou neessarily evaluaing heir deerminan values. By some rules for seleing he design poins o go ino he design, non-promising designs wihin a given design lass are eliminaed and by proper use of he algorihm he number of lasses o be eamined for a pariular problem is redued o only a few. The proedure moves sequenially in he direion of inreasing value of deerminan of informaion mari as he searh proeeds from one opimum o anoher. The required ea D-opimum design is reahed when here is no furher improvemen in he deerminan value of informaion mari. 3. Mehodology In order o solve he problem defined in seion. above he suppor poins are arranged ino H onenri balls, g, g,, g H, suh ha he h h ball, g h, is defined by

5 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG 37 g h h h h nh ; hi = (,,, n ) and onsiss of n h suppor poins and n h = ; h =,,, H. Moreover, eah suppor poin in g h is of ' disane, r h = h i hi ; i =,,, nh from he enre of X ~. Besides, r > r > > r H. The purpose of he grouping is o mae i easy o idenify ses of p p informaion maries wih equal diagonal elemens, so as o ompare he informaion maries wihin a given se (wihou neessarily evaluaing he deerminan values) on he basis of he off-diagonal elemens. 4. The Algorihm The Hill-Climbing ombinaorial algorihm is embodied in he sequene of seps following. If is he H- uple of suppor poins a he h sep, hen he H-uple of suppor poins a (+) s sep is formed by holding H- of he r h values fied and alering he values of jus wo balls. Tha is, only wo values of he r h are alered while he remaining H- values are held fied subje o r h = ; is he design size. Given suppor poins in X ~ whih have been pariioned ino H groups (balls) as, we suppose ha a he h sep of he sequene an H-uple, = {r, r,, r H }, of suppor poins are seleed from he balls. Then he number of available designs a his h sep is H a a h where a h is he number of sub-designs in he h h ball and is ompued simply as nh a h ; nh rh rh However, when n h is less han r h we shall ompue a h as nh a h f h where f h is a posiive ineger value defined by f h = r h - n h and is a posiive ineger value suh ha f h < n h. For onveniene, we shall presen he deails of he algorihm saring wih H =, H = 3, e. S Searh (H=): The S searh is embodied in he sequene of seps in Table below. Table : Combinaoris for Choosing D-Opimal Design Sep Ball ombinaion umber of available designs, a Deerminan Value, d g g r r a d r + r - a d r + r - a d m r +m r -m a m d m m+ r +m+ r -m- a m+ d (m+) m+ r - r + a m+ d (m+) m+3 r - r + a m+3 d (m+3) q r -w r +w a q d q q+ r -w- r +w+ a q+ d (q+) h

6 38 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG - w = q m 4. d < d < < d m > d (m+) d (m+) < d (m+3) < < d q > d (q+) 4. d pp ( i, j) (, j) ma de M ( ) ; M ( i ) a pp S ; =,,,, q+ S is he spae of non-singular pp informaion maries a he h sep. The sequenial seps involved in seing up he able are as follows: ) A =, define an iniial -uple, = ( r, r ) of suppor poins suh ha r, r and r + r = ) Compue he values of a and a designs from ball and ball, respeively, and se a = a a ; a is he number of all he available designs a sep =, 3) Epress he a designs from g and a designs from g, respeively, as follows: () ( a ),,, () ( a ) ;,,,, 4) Define he a designs as omposie designs; i.e. ( a) (,) (, a ),, ( a) (,) (, a ),, () () ( a) ( a,) ( a, a ),, ( a) ( a) ( i, j) pp Where, M S ; i =,,, a ; j =,,, a I will be noied ha he a design measures are grouped ino a ses, eah se onaining a designs and wihin eah se he diagonal elemens of informaion maries are ealy he same. For eample, onsider an eperimenal area whose suppor poins are grouped ino - g = ; g = - ; g 3 = ( ) Suppose = {3, 3, } hen a = 4, a = 4, a 3 =, a = 6 The a = 4 designs from g are - () (3) (4) = - ; = - ; = - - ; = and he a = 4 designs from g are

7 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG 39 - () (3) (4) = - ; = - ; = ; = The four ses of designs are as follows: Se : - = - ; (,) = - ; (,) = - - ; (,3) = - - (,4) Se : Se 3: - = - ; = - ; = - - ; = - - (,) (,) (,3) (,4) = - ; = - ; = - - ; = - - (3,) (3,) (3,3) (3,4) Se 4: - = - ; = - ; = - - ; = - - (4,) (4,) (4,3) (4,4) Using he quadrai model f(, ) = a + a + a + a + a + a +, he informaion maries of he four designs in he firs se are respeively, M = ; M =

8 4 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG - M 3 = ; M 4 = I is learly seen ha he diagonal elemens of he informaion maries of all design in he firs se are ealy he same. This is also rue for he oher designs in he same ses. 5) By omparing he absolue values of he off-diagonal elemens of he informaion maries ( i) belonging o he same se, idenify he bes design,, in he i h se, suh ha ( i) ( i, j) M = ma ( ) j ( i) ( i, j) M ; i =,,, a ; M, M 6) Define suh ha ( i) de {M( )} = ma de{ M ( )} = d ; i =,,, a. i pp S. 7) Se = ( r +, r - ) ; =,,, m+, m+,, q+ and following seps o 5,above obain { d }. From equaion 4., deermine d m and d q. 8) Se d = ma {d, d m, d q }. Then, he orresponding design measure is he required D-opimal ea design. S 3 Searh (H = 3): As wih S searh, we assume ha he suppor poin in X ~ have been grouped ino g, g and g 3 balls and we proeed wih he following seps; ) A =, define an iniial 3-uple = ( r, r, r 3 ) of suppor poins aen from g, g and g 3, respeively, suh ha r, r, r 3 and r + r + r 3 =. ) Holding ball g fied a r, apply he proedures of S searh on he remaining wo balls and obain d 3 (r +) and he orresponding uple ( r, r, r3 ) ; r and r 3 are he opimal number of suppor poins aen from g and g 3, respeively, when ball g is held fied a r. 3) Se =,,, q+ and obain { d3 ( r ) } as in seps and above. Hene, deermine d3 m ( r m ) and d3 q ( r q ). 4) Se d = ma { 3( r ) required D-opimal ea design. S H Searh (for general H) : d, d3m( r m), d3 q ( r q ) }. Then, he orresponding design measure is he For S H searh, he suppor poins in X ~ are arranged ino H balls, namely, g, g,, g H following he usual proedure and he searh proeeds along he following seps; ) A =, define an iniial H-uple, = (r, r,, r H ), of suppor poins aen from g, g,, g H, respeively, suh ha r, r,, r H and r + r + + r H =. ) Holding ball g fied a r, apply he proedures of S H- searh on he remaining H- balls and obain d H (r +) and he orresponding uple, ( r, r, r 3,, r H ) ; r, r 3, r H are he opimal number of suppor poins aen from balls, g, g 3,, g H, respeively, when ball g is held fied a r.

9 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG 4 3) Se =,,, q+ and obain { d H ( r ) } as in seps and above. Hene, deermine d Hm( r m) and d Hq( r q). d = ma { H ( r ) 4) Se d, d Hm( r m), d Hq( r q) }. Then, he orresponding design measure is he required D-opimal ea design. 5. Properies of S H Searh Every sequenial ehnique an generally be haraerized by how o begin he searh, in wha direion o oninue, a wha sep lengh and how o end he searh. The S H searh is governed by erain properies whih for simpliiy in presenaion are oulined below for H =. ) Saring poin: The S searh ommenes a an arbirary -uple of suppor poins = ( r, r ) ; r, r ; r + r = ) Direion of searh: The searh moves in he direion of inreasing values of deerminan, d, as in equaion 4.. The direion is deermined afer eamining all he available designs a he preeding sep. I should be noed ha alhough a % searh is required a eah sep, by he appliaion of Theorem and he properies of he searh, he number of deerminan evaluaions a eah sep redues o no more han a. 3) Sep lengh: The searh moves one sep a a ime in boh he inreasing and he dereasing values of ( r, r ). Tha is, a sep, we have he uple, = ( r +, r - ) or = ( r -, r + ) ; =,, 4) Sopping poin: The searh erminaes a = q+, as in equaion 4.. 5) The se of seps given by { } = { (r, r ) } defines a pah or direion of searh ha is in aordane wih propery above. The se is ompleely ehausive of all possible pahs. In oher words, for any oher saring poin, say, where ( s, s ) ; s = s ; s r s = s ; s r, s, s ; s s = he pah defined by { } and { } will oinide somewhere along he searh. The proof of propery 5 is simple sine any of he omponens in = (r, r ) aes any ineger value from o and eah number pair is suh ha he omponens sum upo. I is obvious ha he ombinaions ehaus all pairs whose omponens sum upo. Consequenly, { } is ehausive of all possible pahs. We show in Theorem ha d is he global D-opimum. I is suffiien o esablish his for H=. Theorem Le d = de M( ) = ma X ~ {de M( )} M( ) be he global value of deerminan of all pp informaion maries for all non-singular designs. Then de M( ) = de M( ), where de M( ) = ma {de M( )} ; =,,, m, m+, m+,, q, q+, S pp

10 4 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG - d = Proof (By Conradiion) Suppose { } onverges o he value d say, ( s, s ) ; s, s ; s s = d, le i be possible o sar a fresh sequene a anoher poin, g g whih onverges o he value d suh ha d = } mus oinide wih { } somewhere along he searh. Then d = d and measure, is he D-opimal ea design. Q.E.D. d. Then wih respe o propery 5 of he S searh, {, he orresponding design 5. umerial Illusraions We presen some illusraions o demonsrae he woring of he algorihm developed in seion 4. The demonsraions are based on firs and seond order models. The essene of demonsraing wih firs order models is o show ha his new approah performs redibly well even for firs order models, and hen eend i o seond order models. However, his does no prelude he woring of he algorihm for higher order models. Illusraion Given a firs order model, f(, ) = a + a + a + a +, defined on a regular geomeri area having a finie number of suppor poins as in figure I, we see o onsru an -poin D-opimal ea design measure, ; = p, p+,, p (-,) (,) (,) (-,) (,) (,) (-,-) (,-) (,-) Figure : A regular geomeri eperimenal region a speified values. By arranging he suppor poins as desribed in 4, we form he following groups - g = ; g = - ; g 3 = ( ) The problem of onsruing D-opimal designs for he model is grealy simplified by he fa ha o ahieve D-opimaliy, eah i mus be a vere poin. A proof of his has been given by Bo and Draper (97) for he ase when =p (i.e., when he design size is he same as he number of parameers in he model) and also by Mihell (974) for he ase when >p. Sine he D-opimal design mus onsis enirely of orner poins (vere poins) we resri ourselves o he vere poins. In his eample he

11 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG 43 vere poins refer o suppors poins in ball g. The design poins of he D-opimal measure, ; = 4, 5, 6, 7, 8, are summarized in Table below. Table : Design Poins for D-opimal Ea Design for a firs order model defined on Figure Design Ball ombinaion Design Poins Deerminan Size, g g g 3 Value 4 4 -,, -, ,, -, --, ,, -, --,, ,, -, --,, --, ,, -, --,,--,-,-. These resuls indiae ha for firs order models defined on a spae of rials as in Figure, and for a given -poin D-opimal ea design measure,, he + poin D-opimal ea design measure, +, is obained by adding a vere poin (in aordane wih he rules for maimizing informaion mari) o. Illusraion Given he firs order model, defined on an irregular spae of rials as in Figure below, we see o onsru -poin D-opimal ea design measure,. (, ) (-, ) (-½,) (½,) (,) (-, ) (,) (,) (-,-) (-) (,-) Figure : An irregular-shaped eperimenal region a disin poins By grouping he vere poins aording o heir disanes from he ener of X ~, he following groups are formed; g = - - g = g 3 = - g 4 = -½ - ½ The operaions ha lead o a 4-poin D-opimal design are as abulaed in Table 3 below.

12 44 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG - Table 3 Combinaoris for obaining a 4-poin D-opimal ea Design on he Irregular Geomeri area in Figure Sep Ball ombinaion umber of available g g g 3 g 4 designs, a.56 Bes Deerminan Value, d SIGULAR SIGULAR 5 4 SIGULAR SIGULAR d =. The orresponding D-opimal ea design measure is ¼ - ¼ = - ¼ ¼ The design poins ha mae up he D-opimal -poin ea design measure ( = 5, 6, 7, 8) are summarized in Table 4 below. Table 4: Design poins of D-opimal ea design for a firs order model defined on figure Design Ball ombinaion Design Poins Deerminan size g g g 3 g 4 Value 5 --, -,, -, , --, -,, -, ,--,-, -,, -, ,--,-, -,-,-,,. Illusraion 3 In his illusraion, we demonsrae he effeiveness in he performane of he algorihm for onsruing D-opimal ea designs in blos of unequal sizes. For he purpose of illusraion, we onsider onsruing a 7-poin D-opimal design measure in wo blos of sizes = 4 and = 3, for he bivariae quadrai surfae, f(, ) = a + a + a + a + a + a +, defined on a regular and oninuous eperimenal area suh as in Figure bu wih 5 grid poins. The poins are arranged in he following groups; - - -½ - -½ ½ -½ g = - g = ½ g 3 = g 4 = ½ ½ g 5 = ½ g 6 = ( ) - -½ - -½ -½ ½ ½ - - ½ - ½ -½ ½ -½ - ½ - -½

13 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG 45 The sequenial proedures for onsruing he 7-poin D-opimal design are laid ou in Table 5 below. Table 5: Combinaoris for obaining a 7-poin D-opimal ea design in blos of sizes = 4 and = 3 o of Available Sep Ball Combinaion Bes Deerminan Value d g g g 3 g 4 g 5 g 6 Designs a SIGULAR SIGULAR SIGULAR SIGULAR d = The orresponding D-opimal design measure is - - = - - or = - - or 7 - ½ - -

14 46 Mary Iwundu e al: A HILL-CLIMBIG: ALGORITHM FOR COSTRUCTIG - ξ 7 = Conlusion This wor has suessfully produed a new approah nown as he Hill-Climbing Combinaorial proedure, for onsruing D-opimal ea designs. Resuls obained show ha he algorihm ompares favourably well wih nown algorihms. Referenes [] A. C. Ainson, A.. Donev, The Consruion of Ea D-Opimum Eperimenal Designs wih Appliaion o Bloing Response Surfae Designs. Biomeria, 76 (989), [] A. C. Ainson, A.. Donev, Opimum Eperimenal Designs, Oford: Oford Universiy Press, (99). [3] M. J. Bo,. R. Draper, Faorial Designs, he XX Crierion, and Some Relaed Maers. Tehnomeris, 3 o.4 (97), [4] R. D. Coo, C. J. ahsheim, A Comparison of Algorihms for Consruing Ea D-Opimum Designs. Tehnomeris, (98), [5] V. V. Fedorov, Theory of Opimal Eperimens. ew Yor. Aademi Press (97). [6] T. Jia-Yeong, On he Sequenial Consruion of D-Opimal Designs. Journal of he Amerian Saisial Assoiaion, 7 (976), [7] M. E. Johnson, C. J. ahsheim, Some Guidelines for Consruing Ea D-opimal Designs on Conve Design Spaes. Tehnomeris, 5 o. 3 (983), [8] S. Karlin, W. J. Sudden, Opimal Eperimenal Designs. Ann. Mah. Sais. 37 (966b), [9] J. Kiefer, J. Wolfowiz, On a Theorem of Hoel and Levine on Erapolaion Designs. Ann. Mah. Sais. 36 (965), [] T. J. Mihell, An Algorihm for he Consruion of D-opimal Eperimenal Designs. Tehnomeris, 6 (974), 3-. [] T. J. Mihell, F. L. Miller, Use of Design Repair o Consru Designs for Speial Linear Models. Rep. ORL Oa Ridge aional Laboraory, Oa Ridge, Tennessee, 466 (97), 3-3. [] I. B. Onuogu, Foundaion of Opimal Eploraion of Response surfaes, Ephraa Press, sua (997). [3] I. B. Onuogu, P. E. Chigbu, Super Convergen Line series in Opimal Design of Eperimens and Mahemaial Programming. AP Epress Publishing Company, sua, (). [4] A. Pazman, Foundaions of Opimal Eperimenal Design. Boson: Reidel (986). [5] D. J. Van Shalwy, On he Design of Miure Eperimens Ph.D. Thesis, Universiy of London (97). [6] H. P. Wynn, The Sequenial Generaion of D-opimum Eperimenal Designs. Ann. Mah. Sais. 4 (97), [7] H. Yeh, M. Huang, On Ea D-Opimal Designs wih Two-level Faors and n Auoorrelaed Observaions. Meria, 6 (5), 6-75.