www.ninh-hoa.com



 

Trở về d_bb  ĐHKH

 

Trở về Trang Tc Giả

 

 

 

 

Sơ Lược Về Cch

Ra Bi V Giải Đp SUDOKU 

Gio Sư T Đồng

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Main Menu

 
 

 


VI NT VHỌA KIỂU
CC TH NGHIỆM

GS Tiến Sĩ T Đồng

Khoa trưởng
trường Đại Học Dược Khoa Si Gn
1974-1975.

    

 

 

 

Bi 2 :

 

SOME FEATURES OF THE DESIGN OF EXPERIMENTS

Dong To,   D. Sc.  

 ABSTRACT:

 

 

RANDOMIZATION BY MAGIC SQUARES:

 

I. Blocking technique in full factorial design:

The following example of 26 (6 factors or variables, 2 levels) factorial design in 8 blocks illustrates the principles of confounding using block variables B1, B2 and B3 (2)

Standard Order                       Variables                            Block

Yates's Run #                           1 2 3 4 5 6                            #

-----------------------                           -----------------                                           -----

1                                                       -   -   -   -   -   -                                              7

2                                                        +  -   -   -   -   -                                             2

3                                                             -   +  -   -   -   -                                        1

4                                                        +  +  -   -   -   -                                             8

5                                                        -   -  +   -   -   -                                             4

6                                                        +  -  +   -   -   -                                             5

7                                                        -   +  +  -   -   -                                             6

8                                                        +  +  +  -   -   -                                             3

-----------------------                           -----------------                                             -----

9                                                        -   -   -   +  -   -                                              3

10                                                     +   -   -   +  -   -                                              6

11                                                         -   +  -   +   -   -                                          5

12                                                     +  +  -   +   -   -                                              4

13                                                     -   -   +  +   -   -                                              8

14                                                     +  -   +  +   -   -                                              1

15                                                     -   +  +  +   -   -                                              2

16                                                     +  +  +  +   -   -                                              7

-----------------------                           -----------------                                            -----

17                                                      -   -   -   -   +  -                                              6

18                                                      +  -   -   -   +  -                                              3

19                                                      -   +  -   -   +  -                                              4

20                                                      +  +  -   -   +  -                                              5

21                                                      -   -  +   -   +  -                                              1

22                                                      +  -  +   -   +  -                                              8

23                                                      -   +  +  -   +  -                                              7

24                                                      +  +  +  -   +  -                                              2

-----------------------                            -----------------                                          -----

25                                                      -   -   -   +  +   -                                             2

26                                                      +  -   -   +  +   -                                             7

27                                                       -   +  -   +  +   -                                            8

28                                                      +  +  -   +  +   -                                             1

29                                                      -   -  +   +  +   -                                             5

30                                                      +  -  +   +  +   -                                             4

31                                                      -   +  +  +  +   -                                             3

32                                                      +  +  +  +  +   -                                             6

-----------------------                            -----------------                                           -----

33                                                      -   -   -   -   -   +                                             5

34                                                      +  -   -   -   -   +                                             4

35                                                      -   +  -   -   -   +                                             3

36                                                      +  +  -   -   -   +                                             6

37                                                      -   -  +   -   -   +                                             2

38                                                      +  -  +   -   -   +                                             7

39                                                      -   +  +  -   -   +                                             8

40                                                      +  +  +  -   -   +                                             1

-----------------------                            -----------------                                           -----

41                                                      -   -   -   +  -   +                                            1

42                                                      +  -   -   +  -   +                                            8

43                                                    -   +  -   +  -   +                                              7

44                                                      +  +  -   +  -   +                                             2

45                                                      -   -  +   +  -   +                                             6

46                                                      +  -  +   +  -   +                                             3

47                                                      -   +  +  +  -   +                                             4

48                                                      +  +  +  +  -   +                                             5

-----------------------                             -----------------                                          -----

49                                                      -   -   -   -   +  +                                             8

50                                                      +  -   -   -   +  +                                             1

51                                                          -   +  -   -   +  +                                         2

52                                                      +  +  -   -   +  +                                             7

53                                                      -   -  +   -   +  +                                             3

54                                                      +  -  +   -   +  +                                             6

55                                                      -   +  +  -   +  +                                             5

56                                                      +  +  +  -   +  +                                             4

-----------------------                            -----------------                                            -----

57                                                      -   -   -   +  +  +                                              4

58                                                     +  -   -   +  +  +                                               5

59                                                     -   +  -   +  +  +                                               6

60                                                     +  +  -   +  +  +                                               3

61                                                     -   -  +   +  +  +                                               7

62                                                     +  -  +   +  +  +                                               2

63                                                     -   +  +  +  +  +                                               1

64                                                     +  +  +  +  +  +                                               8

-----------------------                             -----------------                                            -----

 We can see that block #1 formed from Yates' run # 3, 14, 21, 28, 40, 41, 50, and 63 corresponds to the integers in the first row of the following 8-by-8 magic square III :

 

63

14

21

28

40

41

50

3

 

2

51

44

25

37

24

15

62

8

53

46

31

35

18

9

60

57

12

47

34

30

19

56

5

4

49

22

39

27

42

13

64

61

16

43

38

26

23

52

1

7

54

17

36

32

45

10

59

58

11

20

29

33

48

55

6

8-by-8
MAGIC SQUARE  III

It is also the same for other blocks. In summary, blocks #1, 2, 3, 4, 5, 6, 7, 8 derive respectively from rows #1, 2, 3, 4, 8, 7, 6, 5 of magic square III.
                                                             

II. Taguchi's L8 fractional factorial design:

The L8 orthogonal array using four experimental variables and Taguchi's linear graph 1 is described below (8, 9):

 

Standard Order                    Variables                      L8 Selected

Yates's Run #                       D  C  B  A                      (*) Runs

------------------                     ------------                       -----------

1                                           1  1  1  1                           *

2                                           2  1  1  1

3                                           1  2  1  1 

4                                           2  2  1  1                           *

5                                           1  1  2  1            

6                                           2  1  2  1                           *

7                                           1  2  2  1                           *

8                                           2  2  2  1

------------------                     ------------                       -----------

9                                           1  1  1  2

10                                         2  1  1  2                           *

11                                         1  2  1  2                          *

12                                         2  2  1  2

13                                         1  1  2  2                          *

14                                         2  1  2  2

15                                         1  2  2  2

16                                         2  2  2  2                          *

------------------                     ------------                       -----------

The runs selected, marked with an asterisk, in L8 #1, 4, 6, 7, 10, 11, 13, 16 come from the first two rows of the 4-by-4 magic square IV, or the two diagonals of the 4-by-4 magic square V, below:

 

 

 

 

6

11

1

16

 

 

 

 

13

3

2

16

 

 

 

 

 

 

13

4

10

7

 

 

 

 

12

6

7

9

 

 

 

 

 

 

3

14

8

9

 

 

 

 

8

10

11

5

 

 

 

 

 

 

12

5

15

2

 

 

 

 

1

15

14

4

 

 

 

              4-by-4                                                         4-by-4               

         MAGIC SQUARE IV                              MAGIC SQUARE V 

The runs not used in the magic squares could be used in a full factorial design, or they could be used in the case where a non-negligible interaction between BCD exists. Should we have to break L8 runs down into two sub-blocks, the first sub-block would be runs 6, 11, 1, 16, and the second 13, 4, 10, 7.
 

III. Taguchi's L9 fractional factorial design:

A full factorial experiment of four variables at three levels would have demanded 34 = 81 trials. Taguchi determined only 9 treatment combinations as in the L9 (8, 9): 

Standard Order              Variables                    L9

Yates's Run #                  T  A  B  C                    Runs

----------------               ----------                    ----

1                                      1  1  1  1                      *

2                                      2  1  1  1                   

3                                      3  1  1  1

4                                      1  2  1  1

5                                      2  2  1  1

6                                      3  2  1  1

7                                      1  3  1  1

8                                      2  3  1  1

9                                      3  3  1  1

----------------               ----------                  ----

10                                    1  1  2  1

11                                    2  1  2  1

12                                    3  1  2  1

13                                    1  2  2  1

14                                    2  2  2  1

15                                    3  2  2  1

16                                    1  3  2  1

17                                    2  3  2  1

18                                    3  3  2  1                      *

----------------               ----------                     ----                

19                                    1  1  3  1                   

20                                    2  1  3  1

21                                    3  1  3  1

22                                    1  2  3  1

23                                    2  2  3  1                      *

24                                    3  2  3  1

25                                    1  3  3  1

26                                    2  3  3  1

27                                    3  3  3  1

----------------              ----------                       ----

28                                    1  1  1  2

29                                    2  1  1  2

30                                    3  1  1  2

31                                    1  2  1  2

32                                    2  2  1  2

33                                    3  2  1  2

34                                    1  3  1  2

35                                    2  3  1  2                      *

36                                    3  3  1  2

----------------             ----------                       ----

37                                    1  1  2  2

38                                    2  1  2  2

39                                    3  1  2  2

40                                    1  2  2  2                      *

41                                    2  2  2  2

42                                    3  2  2  2

43                                    1  3  2  2

44                                    2  3  2  2

45                                    3  3  2  2

----------------                   ----------               ----

46                                    1  1  3  2                 

47                                    2  1  3  2

48                                    3  1  3  2                      *

49                                    1  2  3  2

50                                    2  2  3  2

51                                    3  2  3  2

52                                    1  3  3  2

53                                    2  3  3  2

54                                    3  3  3  2

----------------               ----------                   ----

55                                    1  1  1  3

56                                    2  1  1  3

57                                    3  1  1  3

58                                    1  2  1  3

59                                    2  2  1  3

60                                    3  2  1  3                      *

61                                    1  3  1  3

62                                    2  3  1  3

63                                    3  3  1  3

----------------               ----------                   ----

64                                    1  1  2  3

65                                    2  1  2  3                      *

66                                   3  1  2  3

67                                    1  2  2  3

68                                    2  2  2  3

69                                    3  2  2  3

70                                    1  3  2  3

71                                    2  3  2  3

72                                    3  3  2  3

----------------                ----------                    ----

73                                    1  1  3  3                   

74                                    2  1  3  3

75                                    3  1  3  3

76                                    1  2  3  3

77                                    2  2  3  3

78                                    3  2  3  3

79                                    1  3  3  3                      *

80                                    2  3  3  3

81                                    3  3  3  3

---------------                  ----------                   ----

 

          The runs selected, marked with an asterisk, in L9 #1, 2, 3, 4, 5, 6, 7, 8, 9 come from the integers 1, 40, 79, 65, 23, 35, 48, 60, 18 in the fourth column of the 9-by-9 quasi-magic square VI: 

 

 

 

21

13

19

79

30

64

29

75

39

 

51

68

11

48

6

59

10

38

78

67

9

77

60

24

47

37

16

32

74

54

55

40

2

17

57

20

49

26

46

12

1

41

81

70

36

56

33

62

25

65

80

42

27

28

8

50

66

45

35

58

22

5

73

15

4

44

72

23

76

34

71

14

31

43

7

53

18

52

3

63

69

61

                                      9-by-9 (-1/+1 error, rows 4-6)
                                                    MAGIC SQUARE   VI 

We can postulate that the Taguchi selection combinations are not perfect, and that any row or column of the following orthogonal 9-by-9 magic square will provide better combinations

5

46

15

56

25

66

35

76

45

54

14

55

24

65

34

75

44

4

13

63

23

64

33

74

43

3

53

62

22

72

32

73

42

2

52

12

21

71

31

81

41

1

51

11

61

70

30

80

40

9

50

10

11

61

29

79

39

8

49

18

59

19

69

78

38

7

48

17

58

27

68

28

37

7

47

16

57

26

67

36

77

9-by-9
MAGIC SQUARE VII

         For example, we can select column 6, which contains Yates' run numbers 1, 18, 26, 34, 42, 50, 58, 66, 74, to randomize 9 runs out of 81 runs.  The resulting nine experiments as randomized by the magic square can now be written as follows: 

                       EXP#     T       A       B       C

 

                             1       T1     A1    B1     C1

                             2       T3     A3    B2     C1    

                             3       T2     A3    B3     C1

                             4       T1     A3    B1     C2

                             5       T3     A2    B2     C2

                             6       T2     A2    B3     C2

                             7       T1     A2    B1     C3

                             8       T3     A1    B2     C3

                             9       T2     A1    B3     C3

 

CONCLUSIONS: 

Examples of using magic squares to randomize factors in design of experiments have been demonstrated. 

The blocking of a full factorial design involving 26 = 64 runs with 6 variables at two-levels using confounding principles of 3 block variables 135, 1256 and 1234 to generate 8 blocks can be done simply by using the 8-by-8 magic square III. 

The L8 orthogonal array of Taguchi's design for a 4 factor experiment at two-levels using linear graph 1, (half of a 24 = 16 runs), is equivalent to the first 2 rows of a 4-by-4 magic square IV or the two diagonals of the magic square V. 

The Taguchi's L9 design for three-level experiments of 4 variables, or randomized 9 runs out of 81 combinations, can be easily found by selecting one column (or row) of a 9-by-9 magic square VII.  Applications of the magic square selection principle could be very helpful to organizing and simplifying results of any dataset. Finally, this gives more choice of combinations of factor-level than techniques involving blocking or linear graphs.

 

Software: Design-Ease, Echip, RS/Discover and RS/Explore, Statistical Analysis System SAS etc....

 

REFERENCES:

 

1) Box, G.E.P., Draper, N.R. (1987): Empirical Model-Building and Response Surfaces, John Wiley & Sons, NY. 

2) Box, G.E.P., Hunter, W.G., Hunter, J.S. (1978): Statistics for Experimenters, Introduction to Design, Data Analysis and Model Building, John Wiley & Sons, NY. 

3)  Montgomery, D.C. (1997): Design and Analysis of Experiments, Fourth Edition,  John Wiley & Sons, NY. 

4)  Ross, P. (1988): Taguchi Techniques for Quality Engineering McGraw-Hill, Inc, NY 

5)   Kurosaka, R.T. (1985): Magic Squares - Byte, 10:383-388 

6)  Reiner, B.S. (1981): Magic Squares and Matrices, The Mathematical Gazette, 81: 250-252  

7)   Sonneborn III, H. (1988):  Magic Squares and Textile Designs,  Access, 7: 10-16 

8)   Ross, P. (1988): Taguchi Techniques for Quality Engineering, McGraw-Hill, Inc.  NY                  

9)  Taguchi, G. (1987): System of Experimental Design, Vol 1, American Supplier Institute, Inc,   Dearborn, MI

 

 

 

 

Gio Sư T Đồng
Khoa trưởng
trường Đại Học Dược Khoa Si Gn
1974-1975.

 

 

  

 

 

www.ninh-hoa.com