反應六標準差之多品質特性混合實驗最佳化演算法

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Optimization of Multi-Response Mixture Experiments

to Achieve Six-Sigma Quality Level

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Optimization of Multi-Response Mixture Experiments to Achieve

Six-Sigma Quality Level

Student

Bo-Ling Wang

Advisor

Lee-Ing Tong

Chung-Ho Wang

!

"

#

A Thesis

Submitted to Department of Industrial Engineering and Management

College of Management National Chiao Tung University In Partial Fulfillment of the Requirements

For the Degree of Master of Science In Industrial Engineering June 2005 Hsin-Chu, Taiwan Republic of China

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!

(Design of Experiments, DOE) !" # $ % & ' () * + ,- ./012 345 6 47 8 $ 9 : ;< => (Mixture Experiments)? @, 5 A B CD E F G H I ,.1> 9 J K L M NOP (Q R> S (component) TRH I (T S )U VW ,XY 0> ,Z [F G H I \ ] ^ _.`a 1b c d e f g [h i j k lm n o p (q rs ,t uv v k * w x ;y zZ({ | 5 } t ~ o O (desirability function) > ,; z Z{ | 5 (A B. Ribardo[ Allen[14]o w x [P , ( u \ (desirability function for

achieving six-sigma quality)! o - > ,;y zZ 5 ( ` u \ NO u \ , . ¡ S ¢ £ ¤ ¥ ¦ § o ¨ ,© ª « ¬ ® ,Y ¯ o ,° ± Z[ Z.

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Optimization of Multi-Response Mixture Experiments

To Achieve Six-Sigma Quality Level

Student

Bo-Ling Wang Adviser

Lee-Ing Tong,

Chung-Ho Wang

Department of Industrial Engineering and Management

National Chiao Tung University

Abstract

Design of experiments (DOE) is useful in finding the optimal parameter-setting efficiently with low experimental cost. However, in some specific areas, engineers often misuse factorial design on mixture experiments. In mixture experiments, the factor (which is also called component) affects the response by its proportion in all components, but not the actual volume. This is the main difference between factorial design and mixture experiments. In addition, simultaneously optimization of multiple responses is increasingly essential. Therefore, this study proposes a procedure to optimize multi-response in mixture experiments. The desirability function of six-sigma quality introduced by Ribardo and Allen [14] is employed in the proposed procedure. The desirability function considers the estimated mean and variance of response

variable and estimate the response under assumptions required in the six-sigma quality. A rubber bowl’s experiment with four components and eleven response variables are utilized to demonstrate the effectiveness of the proposed procedure.·

·

Key Words: Mixture Experiments, component, multi-response, desirability function for achieving six-sigma quality

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1: ;¸ (¹ º ,s » ¼ ½ ¾ ¿ À9* Á Â (Ã Q Ä ( ÅÆ Ç È É Ê Ë Ì Í [Î Ï ÐÌ Í ÑÒ d Ó Ô Õ Ö× Ä ÅÆ [Ø Ù ÚÄ ¾ Û ¼ Ü 9Ý Þ ß: ;à á (â [ã ä å æ Á Â ÑÒ ç è .` a Á Â ÑÒ é ê ë ì í î Ì Í [ï ð 9Ì Í ñ òó ¼ Ö× å æ ;(Ø Ù Úó ¼ ô # Û . £ ¤ Ñõ(|ö 9 _÷ ø 1ù ú û Á Â ù (ü ý þ 4 4 4Ì 4 4 4 4 zÐ _÷ y ù ( ì 4 Ð Î 1» Ñõ{ ú û ( _þ - > 4 4 $ Ð ¼ s : ; ! (" # .$ * Á Â ù (Ý % & ' ( 4£ ) Ð* +,o ¨ Ä : ; ÐÝ $ -(. m [¹ º Ý / 0 1 1Ä 1 k * (|2 Ö × Ä 1 (º ' $ ° 3 (_y 4Lucky445 46 5 47 8 4Red RainÐhipple d ¼ Ä (9 q : ; ø 9 < = > Ä Â Â _þ . ?`,a @A B C D E F G d HI ' ¹ º Ä J K à á ( ó ¼ ,L ¾ Û z] 1`Á Â C F G (¹ M . ¾ 1ÑõN O P Q Í Ý Ò 0Û Ä R S õ(lÝ T U V W (X Y Z [ ,) 1L " \ 1`ó ¼ (- ] ^ _ =Ä (U V W [ þ ¸ 1 `a Ä (Á b. 91B c - - o ¹ º û Ä (¸ 1`òÄ (è % 4 þ ¸ d ý 1 d ¼ 1Ý $ Ð Ref Ö× Ä (. m [g h ij `a Á Â ,k. Î l m n o = p q _Ý 6 $ 6 [r 2005õ7s4t

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9 u * Ë u * o Â ... t v wt v `t v x- y z ¼ 1.1 { | [} ~...1 1.2 t (...2 1.3 e...2 1.4 ...2 x y 2.1 > , [9]...3 2.2 > ,å ³ ...9 2.3 ; zZ ,å ³ [2][13] ...10 2.4 u \ , [7][14] ...10 2.5 ; zZ> ,å ³ ...12 x4y e 3.1 > , ...14 3.1.1 6 ...14 3.1.2 ...14 3.2.1 RNOP 1R s, P ...15 3.2.2 RNOP 1R s, ...15 3.2.3 R ,V ...16 3.2.4 V NO f ...16 3.2.5 e...16 3.3 g ± ð Z ¯ Z...17 3.3.1 ð Z ...17 3.3.2 ...17 x6 y Y¯ 4.1 Y ...19 4.2 ...20 4.3 S ¢ ...21 4.3.1 RNOP 1R s, P ...21 4.3.2 RNOP 1R s, ...22

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4.3.3 R ,V ...24 4.3.4 V ,NO f ...24 4.3.5 e...28 4.3.6 ð Z ...28 4.3.7 ...29 4.3.8 Y¯ ¼ ...30 xy ¼ 5.1 ¼ [^ ...31 5.2 ...32 w v

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w2-1 4 S > [4H I ...4 w2-2 4 S > ...4 w2-3 {3,3} ...5 w2-4 3- S 9 ...5 w2-5 s¡ S ,¢Z£ ¤ ...6 w2-6 d ¥,° ± ...6 w2-7 (q=3, n=2)¦§ P ,> ( 9 )...8 w2-8 u \ w¨ ...12 w3-1 ; zZ> § w...18 w4-1 © ç 5 ª ,NO f 8« ¢w...26 w4-2 © § } º ª A,NO f 8« ¢w...26 w4-3 © § } º ª B,NO f 8« ¢w...27 w4-4 © ¬ ª ,NO f 8« ¢w...27 w4-5 \ ,D ® ~w...29 w4-6 ¯ ° ò \ ,± ² w...29

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`2-1 2×2³ H I ...3 `2-2 RH I ,XY...3 `2-3 u \ ,´ `kµ …...12 `4-1 © ª « ¬ ® , u ...20 `4-2 « ¬ ® ,> `...20 `4-3 NOP H2[H3, A ...21 `4-4 NOP H1, [P ¶ ...22 `4-5 NOP H11R s, ...23 `4-6 À· NOP 1R s, ...23 `4-7 R s,V ...24 `4-8 V Û ¸ ¹NO f ,ANOVA`...25 `4-9 V º 5 ¹NO f ,ANOVA`...25 `4-10 e¯ ° `...28 `4-11 e A ...30 `4-12 e [¯ ° X» `...30

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1.1

1% & b¼( $ 9 [ !o p $ % & ' ,) * + . `a 1b c d e f g [h i j k lm n o p (q rs ,½ ¾ ú , ¿ ,o « !{ |Ú;y zZÀ h i j ,k l w x ; , 5 ½ Á Â [ " # $ % & ' .

¢a r Â (off-line quality control)Ãí ¢ o p H ` g ± e Ä T!¢a r Â ,) * Å B.G !Æ Ç ((Design of Experiments, DOE)[N O f (Response Surface Method, RSM) ° ¹ º 6 ¸ ì 1½ ¾ ( 9 È É " # $ % & ' .Ê õ$ ËÌ Í O [NO f ÀÎ ÏÐ RËÑ .

/0 1: ;2 345 6 7 8 $ 9 D E F G H I l e }CL K ß¯ Ò , ÀÓ H !» Ô $ À , [F G H I , d { !> (Mixture Experiments),? @.Õ > Q Å,NOP (response variables, T zZ)Ö J K Ð H I ,¦W ;× L M 0Q [RH I U Ø V¦W ,XY³ .1 > 9 æ ,H I XY!t ( Ø H I 1> 9Ù ! S (component) 0å ò=F G ,H I 1> 9 ( Q R S U VW , S XY.> ,Z [F G H I \ ] å æ _ ¼ z] ñ ò> g ± . `a i= Ú ; zZA B(k l_!" # Ê õÛ /Ü Ý : ; ; zZA B, }1> ( 9 Þ ;Q ¼ - zZ,A B Dabbas8¸ [10]1 ¼ ß> Z ,÷ Æ à á 6 ¥( 5 - O (desirability function) ; zZ,A B } !å òÅu0Ö â òÅu L J K !$ Ë1 -, ã ¼ ò`o g e.

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ä Ü u \ (six-sigma)å G 1D æ ç è é ê (Motorola) Î ë ± : ; ì $ Tí l u \ , !uî o p $ % & ' ï`a 1[ð ñ 4 ò ó c 8 $ 9 1ô ³ 8H õ w W ,s ö X- ÷ $ * l« : ; 0 u \ å G 1`Ô $ 9Ïô ø ) * .H ` ¼ ùRibardo[Allen[14] o , u \ (desirability function for achieving six-sigma

quality)O=; zZ> 5 ,A B- NO f l S XY o ¨ ß u \ Z â òÅu, e.

1.2

ú =t ~ û L Û ¸ ,; zZ> 5 ã ¼ , Ã* t (!ñ ò> A B Ü Ý - ; zZ, 5 .Ø !ß u \ ,; zZNO f ¯ ° 0Ø l ,> e° - ß u \ Z,â òÅu ° o ¨ 6 è !Ào ü , w .

1.3

¼ u \ ùR s(;y zZý ° ß u \ ,V ùR , u \ V Ó > ,NO f .¡NO f e þØ eùß- y u \ ,V ° = u \ , .

1.4

¼ S y x- y!z ¼ ¼ , } ~[t (ïx y ¼ ,å ³ > 4; zZA B u \ ,å ³ ïx4y ¼ Ø ,; zZ> 5 § ï x6 y!Y¯ © ª ª « ¬ ® , Y ¯ ¼ o ,e ° ± ïxy¥!¼ , ¼ [Ø Ù.

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y > Àæ N aå ³ K sQ ; zZ ,å ³ Ú, u \ ¡¥Q ; zZ> ,å ³ .

2.1 [9]

@{ x- y { | [} ~ 9¥ H I À > ,zZ | kT» P ´ `(Q S XY ¥OÚ> 0Ö F G , H I . `2-1`2-2!Y[9] A4BÑP !ß> S zZ,P j Ú³ H I @`2-1 ù`2-1£ ÑP ,X Y @`2-2.¥À9³ H I 9, (A=25ïB=225)Ð(A=30ïB=270) , S XY -Q å { ( Àß> S ,zZ d { XY,ÑP S ½Q L M NOP (Ó H ¥OÚ> 0Ö ³ H I . A B 25 30 A B 25 30 225 225 1.0: 9.0 1.2: 8.8 270 270 0.85: 9.15 1.0: 9.0 ä P ß> S zZ ¥ µ x_i(i=1 ,2 , ,q)!Ø S U > VW , XY x_ißszZ q i x_i 1, 1 ,2 , , 0≤ ≤ = (2.1) 0 . 1 2 1 1 = + + + = = q q i i x x x x (2.2) i=(2.2),¡Â Úä P ,° ± - y ú .4 S ,> (q=3)!Y w2-1(e Q F G 4H I ( 0> ` 2-1 2×2³ H I ` 2-2 RH I ,XY

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1" # ¡Â x₁+x₂ +x₃ =1.0(q rs £ ¤ w9e 9( 4 f @w2-2. 1> 9 q D Q Ø R S =NOP ,NO f ! e,e.¥ñ ò- µ , S P Ø (NO f s 4 * (1) * æ ,NO f ï(2) Ø ,ð Zï(3)À ¡ , 3 æ , Ø À ,ð Z[9]. 1` µ NO f @s NO ) , , , (x1 x2 xn φ η= (2.3) 1NO f 9 Ú, !x_i(¢Z;, @(2.4)(2.5) = + = q i i i x 1 0 β β η (2.4) ≤ = + + = q j i q j ij i j q i i i x x x β β β η 1 0 (2.5) i=(2.2),¡Â ° ù- _ 9! " 5 ! 0 @ + + + = ≤ ≤ ≤ ≤ = q k j i q k j q k ijk i j k q j i q j ij i j q i i i x x x x x x β β β η 1 (2.6) `a ÀÏ° # 1¦x_i −x_j(i≠ j)(P å .V0$ , (2.3)° k l% æ , . (1,0,0) (0,0,1) (0,1,0) (1,0,0) (0,0,1) (0,1,0)

w 2-1 4 S > [4H I [9] w2-2 4 S > [9]

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s° ± & ' ,Û ¸ [( S Ñ) q ¨ @* æ ,NO f . Û ¸ ,° ± & ' ¥ÚD + ,

{ }

_{q ,}_m [16](Simplex

Lattice Designïm Åxi S XY,S V )q, S 9 [17](Simplex

Centroid Design) .w2-3Q ù - S ² = S ,° ± & ' ( ? . N ïw2-4¥Q 1 S ,° ± & ' (/ [ 9 -0 1 À å òO,NO f .1 æ 3l Rβ 2 ¡ T° K ß- > S ,NO f . `a x_i,? . 1¥ z3 4 sÖ Ü 0ß1 0Q 5 (2.7), q i U x L_i _i _i 1, 1 ,2 , , 0≤ ≤ ≤ ≤ = (2.7) (2.7)9,L_iÐU_iS ] !x_i,s¡[-¡ Àq D Q j 6 $ ã ä â Ú ( æ ¥ S XYx_i7 10~L_iU_i ~1,|d L g ± 8q r Ã 89 Ú(2.7) !x_i(° ± ? . . `| ° : * i x 4x_i#!x_i¢Z£ ¤ ¡,£ P . 0≤L_i ≤x_i ≤1 T S X Ys¡,¡Â ¥: L L x x i i i ₋ − = 1 * _° _Ú₀_≤ * _≤₁ i x À9 = = q i i L L 1 @w2-5ï 0≤x_i ≤U_i ≤1 T S XY-¡,¡Â þ min 1 1 ≤ − = i i q i i U U ¥: 1 # − − = U x U x i i i ° Ú0≤xi# ≤1 À9 = = q i i U U 1 . w 2-4 3- S 9 [8] 1 x (1,0,0) 3 x (0,0,1) 2 x (0,1,0) ( 3 2 ,0, 3 1 ) ( 3 1_, 3 1_, 3 1₎ 1 x (1,0,0) 3 x (0,0,1) 2 x (0,1,0) (0, 2 1_, 2 1₎ ( 2 1 ,0, 2 1 ) w 2-3 {3,3} [8]

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/0 1L g ± ¢Z£ ¤ 0ÚÀ£ S ,? . !0ß 1| Ú° ± 5 d ¥; 0Ö Û ¸ @w2-6ïQ # 1À· ¡Â @ 0 1 ≥ = q i i x H | À9_H∈ q.¥`| 9 L Ú 8 9 < = > º (Computer-Aided Design) . ? @ (< = > º Q 1 NO f ¡ 1° ± ¥ Ó ¥A æ , . ¥ > NO f !_y =_x + À9 ) , ( ~ 2 n n n iid N 0 σ I ¥iF e , _ 2 ,¶ ˆÏB Ü D ® ) ) ( , ( ~ ˆ 2 1 1 − + σ x'x p N (p≥q) 0_yˆ ₌_xˆ ~ _N(_x ,_σ2(_x'_(x'_x)−1_x))_D _® _d ¥ˆÐ_yˆ,P C D Þ !_σ2₍_x'_x₎−1_Ð_σ2_x'₍_x'_x₎−1_x_.D ₊ _,< ₌ _> _º 6 ) 1. F 5 ˆP C D _σ2₍_x'_x₎−1_,± _!t _(,_D- ₅ _ÏT F 5 det(_x'_x)−1_. _\ _À _D _® _¥`Ï° _F ₅ _{_} ₂ ˆ ,E F G & ( H . 1 x 2 x x 3 1 * 1 = x 1 * 2 = x * 1 1 = x w 2-5 s¡ S ,¢Z£ ¤ 1 x 2 x x 3 w 2-6 d ¥,° ±

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2. F 5 ˆP C D _σ2(_x'_x)−1_,ò _¢Ð!t _(,_A- ₅ _ÏT F 5 _tr((_x'_x)−1)_.Àt _(1=F ₅ _{_} ₂ 0 ˆ b = , P . 3. F 5 _yˆ,_P !t (,G- 5 : x x x' x' 1 2 ₍ ₎− σ = = pp p p c c c c c c c C 0 11 10 0 01 00 c_ii∈ i=0~p ¥T!F 5 max(c_ii i 0, ,p) i = . 4. F 5 c_ii!t (,V- 5 . - 6 ) D- 5 D Ð Ú.} -1{ - NO f s À9- ) , A 1À· 4) 9 Ï Ê I =Ø 4) , A [9]. > ,) 1=> S , }d `P ß> S zZ.1> 9 Ï° # 1- d Q > S 01Àd { sÞ ò NOP J L M ,P `Ô P Ù ,!§ P (Process Variables).

K ß`Ô A B| ° J K òÓ > g ± J !- y £ k T1Ó ,q S > 9# L n y § P .q=34n=₂!Y À9 1 0≤ x_i ≤ i=1 ,2 ,3 (2.8) 1 ± = l z l =1 ,2 (2.9) 9 !Y ° K (2.10), 3 2 1 123 3 2 23 3 1 13 2 1 12 3 3 2 2 1 1x x x x x x x x x x x x SC β β β β β β β η = + + + + + + (2.10) 0w x Ñ§ P ¥° K (2.11), 2 1 12 2 2 1 1 0 z z z z PV α α α α η = + + + (2.11)

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* > Ñy £ ( _ 2 γ_τλ =α_λβ_τ À9λ∈

{

1 ,2 ,12

}

Ð

{

1 ,2 ,3 ,12 ,13 ,23 ,123

}

∈ τ S ] òO§ P η Ð 9 _PV η ,2_SC .¥Ø £ @s < = = = < = + + + + + = + + = 3 3 2 1 12 2 1 0 3 1 1 2 12 2 1 0 3 2 1 123 3 3 3 1 ) ( ) ( ) ( ) , ( j i j l l ij i j l ij ij i l l i i l i i j i j ij i j i i i x x z z z x z z z x x x z x x z x z z x γ γ γ γ γ γ β β β η 12 ₁ ₂ ₁ ₂ ₃ 123 2 1 123 0 123 z z z xx x l l l ₊ + + = γ γ γ (2.12) À° ± @w2-7 1(2.12)9,¸ Ï° s ` = = < < = + + + + + = + + + = 2 1 123 1 2 3 3 3 1 3 2 1 0 123 3 0 3 1 0 2 1 12 2 2 1 1 0 ) ( ) ( ) ( ) ( ) , ( l l l j i i j l ij i i l i j i ij i j i i i z x x x x x x x x x x x x z z x z x z x x z x γ γ γ γ γ γ α α α α η 2 1 3 2 1 12 123 3 12 3 1 12 x x x x x x z z j i ij i j i i i + + + < = γ γ γ (2.13) +1 +1 -1 -1 z1 z2 w2-7 (q=3, n=2)¦§ P ,> ( 9 )

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À9(2.12)TòOw2-7,M w 0(2.13)òOw2-7,N w Ñj (kµ O å { .P ° ± & ' d ¥ Ï° < = > º =¦§ P ,> .

2.2

Î Q R [1] ò§ S 9T U V ú [À~ W (VOCs[ NO_x),³ 2 g ± T U X W Á Â Y Z , ÀÚ> ,VW L M NOP , Cornell[9] (> -VW .ÀNOP !T U V ú ! - zZ,> A B. [ \ º [5] 4) ~] ª ò< ^ _ < [0 < É ,L M i=» 4) ~ ] ª ,XY!L M _ < [0 < É ,H õ ã À% > e.ÀÏ< - zZ,> A B. C ` a [4] @* ùß> zZ,P b L (₂k₎_c _é _J _p _`_ÀÚ_XVERT ¡y , Ü 9A- 4D- 4 SNX-" d A 0 L J p `, e f !¡,J p `.Àt ( 1ùß> S ,P 0 L c é J p ` g ± ¡n S ¢ . : g h [6] @* 1ßd Z,Scheffé¢Z4 ¹L p i ,4¹ 9 g ± j k ,D- .l m | [3] @* 1ßd Z,Scheffé¢ Z ¹ 9 g ± j k , A- .À 1 @* " # < = > º , j k Z. Guo8¸ [12] n o 8> S p n q r Z, A À ¦§ P ] s [( t u ÀÛ /{ | y NOP , 5 }Q i v (Trade-Off),eg ± ¼ (d - Q , e. Sandoval-Castro8¸ [15]> 4) w 3x 3,d { XYy X z { | @* L M h 5 ( A (| W .ÀNOP Ïd } - }À t (Ö ` e 0Q iRNOP ,NO f Rw 3 x 3J ,> A . æ aÞ : ;> å ³ , };d Q ; zZ,A B 1`d ~ .

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2.3 [2][13]

1 ; zZ,A B- D + Q 6 $ k+ ! 1L J K e f ,Á Â H I 9 i v ,e e./0 1NOP ;þÐ Á Â H I ,³ 2 ½ ¾ | i v ,ed þÛ /Q 6 $ k+ ¸ 1 Ú Y | Þ / : ;d Z Æ ¹ , e d { .æ NOP ; ¸ { | , Ï¡ , e[ ð , e,\ " q D ^ _. H ` ; zZA B, ¿ , .» J ý ,e!Ö× y P - y ) ` ) - y VNO.1 » y VNO| Ú Ó A B,¦ Ö , ) NO /0 1» ½ ¾ (A B- Ö æ , ) d Q (ó .

1: ;;P W eÐ Ü Ý ¡ Ã Ì Í Ð O1; zZ,A B-.@Ã S S ¢ (Principal Component Analysis, PCA)4 3 (Data Envelopment

Analysis, DEA)4 ë ¼ (Fuzzy Logic)4o ³ ES ¢ (Grey Relational

Analysis)4 Ô I ú ¾ Ä : (TOPSIS)4Ô â (Neural Networks)4 8 ÀÓ ;!1Ö× ) ) ¡ æ , l ° J K e f ,VNO.

% Ribardo[Allen[14]o , u \ ` g ,Åu i å X» , o ¨ - À u \ ,Åu Ú c [ ' å ß u \ ,; zZ, S XY.s!Ø , .

2.4 [7][14]

Q zZNOòÀt u( ú Ø , Q ÅN O1° K ,? . N K Ê t u, ú [7].0 u \ !Ê õr ,) * 1 - Q ùNO, P L w W ÚÀV ß u \ , ° J K i u \ V Ø ;¹) ½ ± ¡,j k Z.Àe @s

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Ribardo[ Allen[14]o , u \ zZP ,Z ° S ! Ë¡[L Ë¡Ñ) sS ] òÀg ± 1. zZP # 1 , Ë¡ zZP # 1 , -s¡ ¶ Ø P NO f , µ [P σ ¡ ¥1 τ =sσ (q rs µ u \ d₍µ_,σ_,τ₎@s

[

( , , ), ( , , )

]

min ) , , (µ σ τ = Y µ σ +τ Y µ σ −τ d (2.14) À9Y(•)`1 % Ò j ,q rs : W τ |,ö À¦ @s + − Φ + − Φ − + − Φ − + − Φ = σ σ µ σ σ µ σ σ µ σ σ µ τ σ µ ) ( ) ( 1 ) ( ) ( ) , , ( s U s L s L s U Y (2.15) (2.15)9,Φ(•)!u D ® S ,¡ H ~ U[L¥S ] ! zZ P , -4s¡. * ° NO u \ , Àkµ T!1 j | 1.5¢ u \ !° K ,E \ s( ö . 2. zZP L , Ë¡ æ zZP d # 1 , -s¡ 0t u[\ ° K !£ W NO: ; u | µ u \ @s

) 5 . 1 ( ) (Y =Φ z_eff − d (2.16) (2.16),Φ(•)!u D ® S ,¡ H ~ 0 − − + = M T M Y z_eff 1.5 3 À 9Y! zZP ,NO4M ! zZP ,\ ° K 4T! zZP ,t u. , UL# 1 , Ud # 1 , Ld # 1

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¥- ¦ ,R zZP ( ¥V D(x) !

[

]

[

]

{

w

}

S m m w r r w r r r d r d m d d D(x) ₁(Y₁(x)) (Y (x)) ₁(Y ₁(x)) +1 (Y (x)) 1 + + = (2.17) À9w_i! ) .1D(x)91ßr!# 1 Ë¡, zZP r+1ß xm !Ãý Ö t u, zZP .1 u \ τ =1.5 (q rs À ,w¨ @w2-8.

`a Ribardo[Alleno , u \ ,´ `kµ @`2-3.

kµ ¤ 0.9999966~1.00 a ß u \ , L 9 g ± 0.9938~0.9999966 a ß6 ß u \ , f ö }Ö ¥ , 0.9332~0.9938 a ß4ß6 u \ , ° K }Þ 0.69~0.9332 a ßÑß4u \ , d ° K , 0.00~0.69 Ñu \ s, Û ³ d ° K

2.5

¦§ ³ =; zZ> ,å ³ å æ ¨ Dabbas 8¸ [10]O > =÷ Æ à ,á 6 ¥ 5 Àt (1= æ XY, © á 6 ¥.Ø Ú ù;y NOP - Åu NO

`2-3 u \ ,´ `kµ [14] w2-8 u \ w¨ 0 0.2 0.4 0.6 0.8 1 0.5 1 1.5 1.5 1 0.5 3 1 = σ U L i µ ) , (µ σ d 6 1 = σ

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f e.Ø w x § P þÀÚ,Åu!å òÅ u d @ u \ ,â òÅu.

Smith[Cornell[18]Ú w(Biplot){ |`NOP [> S P ,³ 2 . wQ Ý zZ,å ³ ZRH I òRNOP , L M A ª ° ¹ º ¸ ì « ¬ Rä P [OP (³ 2 } B e,l .

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ßt ~ !® > ,A BL Ø - Û ¸ , ; zZA B( þH O u \ r ,8* Z ¼ o - ° N O u \ ,; zZ> ( 5 . Ø ,; z Z> 5 § ¦4y ¯ ° .yRF ùÄ R¯ ° , ± ± ². g ± > , ï u \ S ¢ , eï¡g ± ð Z ¯ Z e ,° ± Z.

3.1

F ùÜ 8* ,P ³ ´ ß æ ,> .

3.1.1

6 $ ¸ ì ,â Ú R> S ,° ± & ' § P , NO P ,t ut u& ' 4-s¡° K ? . 4[å ò ) À· å ³ .

3.1.2

ä P ,Z æ ,> 1. S P ßÛ ¸ ,° ± & ' ¥ ° K ,;¹Ú 9 æ , η=φ (x1 ,x2 , ,xn) . 2. S P L â û P £ ¤ ¦Û ¸ ° ± & ' ¥ ° K , ;¹F 9 ÚD- 5 æ , ) , , , (x1 x2 xn φ η = .

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3.2

F @* u \ S ¢ , l eÀV .

3.2.1

3.1.2 , ß Ë¡, zZP ( P À¶ @s ) ( ) ( ˆ_ij x_i =E η_ij (3.1) ) ( ) ( ˆ2 ij i ij Varη σ x = (3.2) À9η_ij =φ(x_i) x_i!xi ¹, e η_ij!xi ¹, eòO=xjy N OP ý ° .

3.2.2

NOP # 1 , Ë¡ þ 1 s_j¢ u \ (q rs 3.2.1 ,µˆ_ij(x_i) ˆ ( ) ˆ2( ) i ij i ij x σ x σ = ° RNOP 1R s, dij(xi)

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(3.4)9,Φ(•)!u D ® S ,¡ H ~ U_j4L_jS ] !xjy zZ P ,-4s¡. æ NOP d ß , Ë¡ # 1t uT_j[\ ° K M_j ¥ RNOP 1R s, dij(xi)! ) 5 . 1 ( ) ( ij =Φ eff ,i ,j − ij Y z d (3.5) (3.5),Φ(•)!u D ® S ,¡ H ~ 0 − − + = j j j ij j i eff _T _M M Y z , , 1.5 3 .· * ß u \ , :s_j=1.5 ¥Ø ° u \ , @`2-3.

3.2.3

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3.2.4

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3.2.5

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3.3

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3.3.1

3.1.2 , g ± ,ð Z .ð Z µ ° S !Ñy ¶ S ,· æ Z , Z.~ j ¦NO f \ ,D ® 48P ¸ Z ý ¹ \ ,D ® ~w[¯ ° ò \ ,± ² wQ ( D ; ® g ± .¡j ¥Q 1 ¯ ° Ã 89 ? Z, À A g ± ú (lack-of-fit test). q û ð Z ¥k g ± 3£ ¤ ) £ ï N, ¥g ± s- ± ².

3.3.2

! K , e1 º |ß Z 89 g ± . ,g ± ¹ù 8H õ Ú , A [¯ ° å \ ñ ; ¥O) £ æ ,ä P [NOP ïN, ¥Ø eß Z ¥Ø e ° ± . , e§ @w3-1.

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« ¬ ® Q © ª ª å G 9() * » i=1Úû 9 â D ßê ¼ [½ ¾ 8a' L M þ% Ò [ ª å G 9,¿ À Á 8Â 3K Ã ÚÀÄ Z_Å . Æ « ¬ ® , e ÚÀ1- Ç È É 9 % | ð D Ü Ê Î . 6 þ â L M « ¬ ® ,Ë ú [Ä Z8Z ,Ã* H õ ! û 9# L (ç Ì ª 4§ } º ª [ ¬ ª 85 Ý 3) Ô [Í W .i= ó Î â æ (5 Ý 3) Ô ã Y,t (1= « ¬ ® û 9 R5 Ý 3( Ï W XY. « ¬ ® , - ³ ´ 89 ù æ XY,« ¬ Ó 3[5 Ý eÏ L « ¬ ~(@ v w- ) T° z e,« ¬ 1 (@ v w ) z ~ Ð ½ ¡ « ¬ ® , . ¡ * Ø e,« ¬ ® Q ( 1 ª å G 9% |ð D º 89 â û - å (° ê .YÑ ê ~(@ v w4)g ± « ¬ 1 ,R) ° `a ! Ò % |Ú1 ª å G 9Ú¡(È É S ] ç 5 Ó (@ v w6 )[Ä Á Ó (@ v w)g ± # É ç 5 Ä [Ä Á # É ç 5 Ä Ñ ê ~ ° Ø « ¬ ® 1% | =Ç 6 È É 9 Q ( Þ ð D º . â Ô ¶ u Õ ,© ª Á ½ ª Ö ,« ¬ ® ° u Y , zZ× ú 4Ø ê Ë ú 4Ù % 4½ ¾ î Ú P ¨ H P 5 8 D ® 4ç 5 ZÄ Á Z8d { q r - zZ° S !R - y NOP RNOP Ë¡[t u@`4-1.

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6 0.167 0.25 0.25 0.333 7 0.167 0.1915 0.1915 0.45 8 0.167 0.133 0.25 0.45 9 0.06 0.25 0.133 0.557 10 0.167 0.133 0.1915 0.5085 11 0.067 0.133 0.133 0.667 12 0.1042 0.1812 0.133 0.5816 13 0.06 0.25 0.133 0.557 14 0.167 0.25 0.133 0.45 15 0.06 0.1915 0.25 0.4985 -`,R ¹9 À9Ñ¹Q ! g ± ¡n , ú 04¹ ) ½ ¥Q ¶ ,Ý E \ ÀI R ¹!D- 5 Ó ¥ ( .1` RNO f η_j =φ_j₍x₁_,x₂_,x₃_,x₄₎.

4.3

° òO¦ e,¯ ° [¯ ° 4.4.3.1¦4.3.5RF ,S ¢ Ä l S < =¯ ° ,§ ï4.3.6 [4.3.7F ¥! ð Z [ Z¯ , A ¥< =¯ ° 4,§ .

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9 2 -7 10 1 -9 11 0 -8 12 1 -6 13 0 -7 14 2 -8 15 2 -8 i`4-3° + â û 6 þ â e , 1NOP H2[ H3( ` Î Þ =j ã 1Y,S ¢ 9 d ñ ò» Ñy P g ± S ¢ .1` k ¶ NOP H11R s, [P .

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1. H1, i=H1# 1 -4s¡ ã ´ L (3.4) × + − Φ − × + − Φ = 1 , 1 , 1 1 , 1 , 1 , 1 1 , 1 , _ˆ ) ˆ ˆ ( 65 ˆ ) ˆ ˆ ( 75 H i H i H H i H i H i H H i H i s s Y σ σ µ σ σ µ (4.2) i= Æ ß u \ , ã :s_H₁=1.5 S ] Y_i_,_H₁(µˆ_i_,_H₁ ,σˆ_i_,_H₁ ,1.5)[Y_i_,_H₁(µˆ_i_,_H₁ ,σˆ_i_,_H₁ ,−1.5) ¥° K ßNOP H11 R s, d_i_,_H₁(x_i) @`4-5. 2. ? H14H2[H3a ÀI S y NOP d ß , Ë¡ # 1 t uT_j[\ ° K M_j H `(3.5) À1R s, d_ij₍x_i₎ A @`4-6. Run Yi,H1(µî,H1 ,σî,H1 ,1.5) Yi,H1(µî,H1 ,σî,H1 ,−1.5) di,H1(xi) 1 0.999999301 0.999817806 0.999817806 2 0.999801547 0.999860928 0.999801547 3 0.950278792 0.999998321 0.950278792 14 0.999999904 0.986918080 0.986918080 15 0.950278792 0.999998321 0.950278792 Run di,E1(xi) di,T1(xi) di,E2(xi) di,T2(xi) di,C(xi) di,E3(xi) di,T3(xi) di,V(xi) 1 0.539431 0.960626 0.997495 0.998005 0.820341 0.919648 0.99214 0.868643 2 0.705401 0.962134 0.686472 0.991388 0.935325 0.947124 0.985286 0.830211 3 0.266314 0.633825 0.620349 0.536477 0.954486 0.910332 0.54425 0.758036 14 0.815141 0.976483 0.990825 0.975851 0.938625 0.948317 0.994466 0.932543 15 0.266314 0.81273 0.950855 0.99441 0.959941 0.977555 0.950855 0.525913 4-5 H1 4-6

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4.3.3

4.3.2 (3.6) D_i₍x_i₎ 4-7 ! " Run Di(xi) 1 0.885679 2 0.885472 3 0.644311 4 0.655295 5 0.935072 6 0.862478 7 0.955683 8 0.955507 9 0.749178 10 0.922825 11 0.608523 12 0.681324 13 0.698211 14 0.949453 15 0.766436

4.3.4

#4.3.3 $ % & ' ( ) * + Design Expert , - . / 0 1 2 3 45 6 7 ANOVA 4-8 ! "

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8 2 3 45 6 7 _R2_=0.9390_9: ₃ _R2_=0.8291_; _< _ANOVA₌ _> _? @ A 45 B C % + DE F G H I J KL M A B C % + D N = O P Q 45 . / 6 7 4-9 ! " 8 P Q 6 7 R S G H x₁x₂Tx₃x₄B C % + DU V 5 DTW G H 4 5 B C % + DXY Z [ D\ & 6 7 R2T: 3 R20 ] ^ 0.9271T 0.8866_ ` R2a b c d e: 3 R2f g 5.5%h ijk + P Q N . / 6 7 . / l m n 4 3 2 1 4 3 2 1 4 3 2 1 83808 . 7 09111 . 9 52044 . 0 65374 . 2 49026 . 2 90092 . 4 ) , , , ( x x x x x x x x x x x x + − − − + = η (4.3) 4-8 2 3 45 . / 6 7 ANOVA 4-9 P Q 45 . / 6 7 ANOVA

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o4-1 p q r Q s . / t V u o 2 x 4 x 3 x 2 x 4 x 3 x o4-19o4-29o4-3vo4-40 ] ^ p q r Q s =16.70%9w x y s A =25.00%9w x y s B=18.68%vz { s =48.65% . / t V u o" o 4-2 p q w x y s A . / t V u o 1 x 1 x 3 x 3 x 4 x 4 x

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o4-4 p q z { s . / t V u o 1 x 2 x x1 3 x 2 x 3 x o 4-3 p q w x y s B . / t V u o 1 x 1 x 2 x 2 x 4 x 4 x

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4.3.5

* + Design Expert| } 4.3.4 \ ~ . / 6 7 l 4-10 ! " r s w x y s Aw x y s B z { s | 0.1670 0.2500 0.1554 0.4276 0.9474 0.1670 0.1332 0.2147 0.4851 0.9420 0.1670 0.2257 0.1719 0.4355 0.9417 0.1670 0.1330 0.2233 0.4767 0.9414 0.1670 0.2453 0.1330 0.4547 0.9413 0.1670 0.2500 0.1844 0.3986 0.9408 0.1670 0.1630 0.1993 0.4707 0.9383 4-10= #₍x₁_,x₂ _,x₃_,x₄₎=₍₀_.₁₆₇_,₀_.25_,₀_.1554_,₀_.4276₎ | 0.9474^ V ^ ij l l X 0.9332# " 4.3.2 \ q X 1.5 [ h 4-10 ^ ~ + [ #RibardoT Allen[14] [ 8 l T ¡ l X¢£ ¤¥¦ [ "

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(4.3) O . / 6 7 § [ ¨ © ª « o(o4-5)T| § [ ¬ o(o4-6) ! 8 4oE ® ¯ ° G ± ¨ © ² 9t ³ v´ µ ² t ¶ "·4-9\ G ! 8 6 7 ~ ¸ ¹ º q p=0.8136» 8 ® ¯ ¼ ½ G ! & 6 7 W ~ " 6 7 ¾ ¿ ² º q ¾ ¨ , - ) À Á Â ? ² ¼ " 4-10 l |

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4.3.7

Ã Ä 4.3.5 Å q l ₍x₁_,x₂ _,x₃_,x₄₎=₍₀_.₁₆₇_,₀_.25_,₀_.1554_,₀_.4276₎, - ¿ Æ 4-11 ! " Ä 4-1\ Ç È É Ê & l XË Ç È É Ê ÌG ! & l ¯ Í Î Ï Ð º Ñ Ç q º "Ò Ã 4.3 6 7 & l Ó ) ( i D x =0.9682¢£ ¤¥¦ [ "· & l | o4-5 § [ ¨ © ª « o o 4-6 | § [ ¬ o

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v 95%Ô Õ Ö × 4-12 ! " H1 68 T1 271.80 E1 165.20 H2 0 T2 0.07% E2 5.08% C 3.23% H3 -8 T3 0.15% E3 5.21% V 2.34% | ¼ 95%Ô Õ Ö × Ê 95%Ô Õ Ö × Ê _0.9474 _0.9682 _0.8884 _1.0000 4-12G ! ¼ T| [ ³ W Ø Ù Ë | 95%Ô Õ Ö × ÌG ! 8 Ú 0 Â ? ² Û Ü Ý Þ Ã 8 l ß à & á â ã â + ä å æ ç "

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5.1

ö ÷ 9Q ø ù ú t ô û ü ý ì þ ' q Q " ¨ Z + » l Ç E l ? » 8 ¨ F O £ | ñ Ú » ^ ' i T » W Ç ï F £ | ñ Ú " _ ` Ì X¯ ' ! " eì Ê # í ² " $ % W & ' , T( ) * + , - . / 0 ô è > * ù ß à 1 2 2 3 4 ì 5 í ² 6 47 8 ( ) * + "9 _ ` > : ý ì ì î e ; < ì í ² Q ' $ F 4 Q ì 5 í ² 4 Ì ! " = #C > l , - l · ? ¯ @ A B + ' ì í ² 4 Q î CDabbas t D [10]E + F G ì í ² " ` · ^ ² C6 + õ S ·F Ó H G "·Ribardo TAllen[14] # [ î ^ I ¯ [ ² "h iè é U J + ' Ý K ) J 2 3 = - ' ì í ² Q î "iè é #0 1 L M @ N O á â ã â + ä å æ ç Ó P j Q ° iè é Ç w m T0 1 À Á E ¼ ¯ ² " ? U iè é R " S 3 n 1. iè é Ý K ' w m = #F G I W Ç T = - ; Ö U ' = # 4ü Ú 0 Tw m ' , - Ç » 8 6 F G Ø ì ' " 2. iè é k + . / V B Ú iW X ù 4× Y Z t 5 ¯ Ê / 0 6 [ + í \ T Ý K 6 7 E ] l " 3. iè é B + [ î l 6 [ V $ ß m ò q ó ô 4 ó ô Û « "

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5.2

iè é ' ì í ² Q î ^ ¯ # _ ` a × n

1. iè é k + RibardoTAllen[14] [ î F ° ¿ Ç È É Ê l / $ Cü b ·F [ = Ò c 4 d E ® ¯ e·$ b ^ Ö × · f q 4F I [ "» 8 iè é Ý Þ g = #h i W j b > : W [ " 2. $ l XÒ 9 [ (V k Ò 9 1)4 [ ³ S l m . / 6 7 \ × ® ¯ G H [ ³ n #, - 0 1 " Ç È É Ê o p ù [ = Ò c q r b s 4t ¯ = 6 > ó u / 0 "» 8 iè é Ý Þ g è é = _ Û [ î N B O Ò 9 [ 4^ ` = #¾ ¨ A B + . / 6 7 , - 0 1 "

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[1] v w x 2001y 0 1 z { T| } H ~ ' H 6 µ Ø w - è é ! " [2] 2003y ì b x 6 ; è é # l ß m ^ j µ Ú Ø ß à ø m è é ! " [3] 2002y ' Scheffé6 7 ò A ~ µ \ Ø + @ è é ! " [4] 2002y ¯ ' » B è é Ø ø û ø m T G @ ! " [5] y 2002y ; ¡ l ¢ £ ² 6 è é µ Ú Ø Q ø m @ ! " [6] ý ¤ ¥ 2002y ' Scheffé6 7 ò D ~ µ \ Ø + @ è é ! " [7] ¦ § ¨ 2004y [ ì í ² 4 Q î µ B © Ø ø û ø m G @ ! "

[8] Cornell J. A., “A Comparison between Two Ten-Point Designs for Studying

Three-Component Mixture Systems,” Journal of Quality Technology, Vol. 18, pp. 1-15, 1986.

[9] Cornell J. A., Experiments with Mixtures: Designs, Models, and the Analysis of

Mixture Data, Prentice Hall, Upper Saddle River, New Jersey, 2002.

[10] Dabbas R. M., Fowler J. W., Rollier D. A., and McCarville D., “Multiple

Response Optimization Using Mixture-Designed Experiments and Desirability Functions in Semiconductor Scheduling,” International Journal of Production

Research, Vol. 41, No. 5, pp. 939-961, 2003.

[11] Derringer G. C. and Suich R., “Simultaneous Optimization of Several Response

Variables,” Journal of Quality Technology, Vol. 12, No. 4, pp. 214-219, 1980. [12] Guo T., Geaghan J. P., and Rusch K. A., “Determination of Optimum Ingredients

for Phosphogypsum Composite Stability under Marine Conditions-Response Surface Analysis with Process Variables,” Journal of Environmental Engineering, Vol. 129, No. 4, pp. 358-365, April 2003.

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[13] Jeyapaul R., Shahabudeen P., Krishnaiah K., “Quality Management Research by

Considering Multi-Response Problems in the Taguchi Method – A Review,” The

International Journal of Advanced Manufacturing Technology, Jan, 2005.

[14] Ribardo C. and Allen T., “An Alternative Desirability Function for Achieving 'Six

Sigma' Quality,” Quality and Reliability Engineering International, Vol. 19, pp. 227-240, 2003.

[15] Sandoval-Castro C. Capetillo-Leal A., Cetina-Góngora C., Ramirez-Avilés R., L.,

“A Mixture Simplex Design to Study Associative Effects with An in Vitro Gas Production Technique,” Animal Feed Science and Technology, Vol. 101, pp. 191-200, Oct. 2002.

[16] Scheffé H., “Experiments with Mixtures,” Journal of the Royal Statistical Society,

B, Vol. 20, No. 2, pp. 344-360, 1958.

[17] Scheffé H., “Simplex-Centroid Design for Experiments with Mixtures,” Journal of

the Royal Statistical Society, B, Vol. 25, No. 2, pp. 235-263, 1963.

[18] Smith W. F. and Cornell J. A., “Biplot Display for Looking at Multiple Response Data in Mixture Experiments,” Technometrics, Vol. 35, No. 4, pp. 337-350, 1993.

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