Optimising a Hesitating Mitchell

Output of the program balans for optimising a Hesitating Mitchell (also called Expanded Mitchell)

The original quality factor is Qf = 54.88
The quality factor after optimisation is Qf = 82.46
The optimised movement can be found in the text but also separately here.

Inserted comments are displayed like this

command line:

balans -r1 -t1 -m 14p8rScrEM.txt 14p8rexpandedM.txt

-r1 : keeps round 1 fixed
-t1 : keeps table 1 fixed
-m 14p8rScrEM.txt : output file
14p8rexpandedM.txt : input file

# option fix round 1
# option fix table 1
12-04-2022 13:28:51
balans v7.62
ensemble size: 4      number of threads: 4
w1=2             w2=1            w1/w2=2             use_Qf1av=1
14p8rexpandedM.txt: 14 pairs, 8 rounds 7 tables, h = 6, S = 336
amount of competition S= 336
number of pairs of pairs N= 91

initial movement scheme:
14  7  8  8 0 g
 1- 8  1   2- 9  2   3-10  3   4-11  5   5-12  6   6-13  7   7-14  8
 1- 7  2   2- 8  3   3- 9  4   4-10  6   5-11  7   6-12  8  14-13  1
 1-14  3   2- 7  4   3- 8  5   4- 9  7   5-10  8   6-11  1  13-12  2
 1-13  4   2-14  5   3- 7  6   4- 8  8   5- 9  1   6-10  2  12-11  3
 1-12  5   2-13  6   3-14  7   4- 7  1   5- 8  2   6- 9  3  11-10  4
 1-11  6   2-12  7   3-13  8   4-14  2   5- 7  3   6- 8  4  10- 9  5
 1-10  7   2-11  8   3-12  1   4-13  3   5-14  4   6- 7  5   9- 8  6
 1- 9  8   2-10  1   3-11  2   4-12  4   5-13  5   6-14  6   8- 7  7

The left matrix indicates the number of times a given board group is played by a given pair
(in this case all pairs nicely play each board group once)
The right matrix is the table of opponent encounters
(in this case: not all pairs meet)

    1 2 3 5 6 7 8 4     1 2 3 4 5 6 7 8 91011121314
 1: 1 1 1 1 1 1 1 1     \ . . . . . 1 1 1 1 1 1 1 1
 2: 1 1 1 1 1 1 1 1     . \ . . . . 1 1 1 1 1 1 1 1
 3: 1 1 1 1 1 1 1 1     . . \ . . . 1 1 1 1 1 1 1 1
 4: 1 1 1 1 1 1 1 1     . . . \ . . 1 1 1 1 1 1 1 1
 5: 1 1 1 1 1 1 1 1     . . . . \ . 1 1 1 1 1 1 1 1
 6: 1 1 1 1 1 1 1 1     . . . . . \ 1 1 1 1 1 1 1 1
 7: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 \ 1 . . . . . 1
 8: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 1 \ 1 . . . . .
 9: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 . 1 \ 1 . . . .
10: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 . . 1 \ 1 . . .
11: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 . . . 1 \ 1 . .
12: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 . . . . 1 \ 1 .
13: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 . . . . . 1 \ 1
14: 1 1 1 1 1 1 1 1     1 1 1 1 1 1 1 . . . . . 1 \

the score matrix:
   *   8   8   8   8   8   1   1   1   1   1   1   1   1
   8   *   8   8   8   8   1   1   1   1   1   1   1   1
   8   8   *   8   8   8   1   1   1   1   1   1   1   1
   8   8   8   *   8   8   1   1   1   1   1   1   1   1
   8   8   8   8   *   8   1   1   1   1   1   1   1   1
   8   8   8   8   8   *   1   1   1   1   1   1   1   1
   1   1   1   1   1   1   *  11   4   4   4   4   4  11
   1   1   1   1   1   1  11   *  11   4   4   4   4   4
   1   1   1   1   1   1   4  11   *  11   4   4   4   4
   1   1   1   1   1   1   4   4  11   *  11   4   4   4
   1   1   1   1   1   1   4   4   4  11   *  11   4   4
   1   1   1   1   1   1   4   4   4   4  11   *  11   4
   1   1   1   1   1   1   4   4   4   4   4  11   *  11
   1   1   1   1   1   1  11   4   4   4   4   4  11   *

The average value in this case is 336/91 = 3.69. A perfect balance is therefore not achievable. In the ideal case, this matrix would consist of only 3s and 4s.

Now successively follow
S2=sum of (score - average)^2,
S4=sum of (score - average)^4,
sd= standard deviation,
d4=4th root of (S4)/N,
and the quality factors Qc, Qf and Qo

sum of squares original:  1055.4
sum of 4th powers: 30501.6
sd= 3.406, d4= 4.279, Qc=54.03, Qf=54.88, Qo=61.54

The following is a table of which tables to keep fixed while optimising.

Fixed tables, indicated by 1
 1 1 1 1 1 1 1
 1 0 0 0 0 0 0
 1 0 0 0 0 0 0
 1 0 0 0 0 0 0
 1 0 0 0 0 0 0
 1 0 0 0 0 0 0
 1 0 0 0 0 0 0
 1 0 0 0 0 0 0


Trying to Improve the Balance v7.62
algo 0: fast temperature fluctuations between T=1 and T=2000
range for Qf1max: 1 - 14
relative weight of Qf1max:          2

During iterations is shown when finding an improvement:
iteration number, temperature, Qf, Qf1av, Qf1max, pair number of this Qf1 maximum, d4, number/fraction of members with best weighted quality
The Windows program balans.exe displays this information in a window.
The Windows program balans3.exe and balans on Unix/Linux displays the information in the console (stdout), e.g. like this (example for system with 4 logical cpus, so by default 4 independent optimisations carried out in parallel = 4 members of ensemble):

   iter      T     Qf   Qf1av Qf1max  pair    d4 nBest
      0    0.0  54.88  52.205  54.39    1  4.279  4/4
      1    1.0  79.95  73.146  77.50   12  2.581  1/4
      3    4.0  80.77  73.977  76.86   13  2.484  1/4
      4    5.0  80.77  74.104  80.87    8  2.461  1/4
      6   10.0  81.61  74.443  78.15   12  2.339  1/4
      8   10.0  82.46  75.347  80.17   14  2.249  1/4
      9   10.0  82.46  75.347  80.17   14  2.249  2/4
     10   10.0  82.46  75.347  80.17   14  2.249  3/4
    104   10.0  82.46  75.347  80.17   14  2.249  4/4
=== Results for all   4 members of ensemble: ======
      #      T     Qf   Qf1av Qf1max  pair    d4
      1   20.0  82.46  75.347  80.17   14  2.249
      2   20.0  82.46  75.347  80.17   14  2.249
      3   10.0  82.46  75.522  78.81   14  2.249
      4   10.0  82.46  75.347  80.17   14  2.249
===================================================

(The solution with Qf1av=75.522 and Qf1max=78.81 happens to have exactly the same weighted quality as the others with the default weight of 2).
Afterwards a table is shown in which we can see which tables are turned (arrow switched) in the final selected solution:

Switched tables indicated by 2, fixed tables indicated by 1
 1 1 1 1 1 1 1
 1 2 0 0 2 0 2
 1 0 0 0 0 2 2
 1 0 0 0 2 0 2
 1 0 2 2 0 0 2
 1 0 0 0 0 2 0
 1 0 0 2 2 0 0
 1 0 0 2 0 0 0


optimized movement scheme:
14  7  8  8 0 g
 1- 8  1   2- 9  2   3-10  3   4-11  5   5-12  6   6-13  7   7-14  8
 1- 7  2   8- 2  3   3- 9  4   4-10  6  11- 5  7   6-12  8  13-14  1
 1-14  3   2- 7  4   3- 8  5   4- 9  7   5-10  8  11- 6  1  12-13  2
 1-13  4   2-14  5   3- 7  6   4- 8  8   9- 5  1   6-10  2  11-12  3
 1-12  5   2-13  6  14- 3  7   7- 4  1   5- 8  2   6- 9  3  10-11  4
 1-11  6   2-12  7   3-13  8   4-14  2   5- 7  3   8- 6  4  10- 9  5
 1-10  7   2-11  8   3-12  1  13- 4  3  14- 5  4   6- 7  5   9- 8  6
 1- 9  8   2-10  1   3-11  2  12- 4  4   5-13  5   6-14  6   8- 7  7

the score matrix:
   *   6   6   2   2   4   3   5   3   3   5   3   3   3
   6   *   4   4   0   2   5   3   5   5   3   5   1   5
   6   4   *   0   4   2   5   3   5   5   3   5   5   1
   2   4   0   *   4   6   5   3   5   5   3   5   1   5
   2   0   4   4   *   6   5   3   5   5   3   5   5   1
   4   2   2   6   6   *   3   5   3   3   5   3   3   3
   3   5   5   5   5   3   *   5   4   0   2   0   4   7
   5   3   3   3   3   5   5   *   5   2   4   2   2   6
   3   5   5   5   5   3   4   5   *   7   2   0   4   0
   3   5   5   5   5   3   0   2   7   *   5   4   0   4
   5   3   3   3   3   5   2   4   2   5   *   5   6   2
   3   5   5   5   5   3   0   2   0   4   5   *   7   4
   3   1   5   1   5   3   4   2   4   0   6   7   *   7
   3   5   1   5   1   3   7   6   0   4   2   4   7   *

Summary of results

sum of squares    was  1055.4, now   287.4
sum of 4th powers was 30501.6, now  2328.8

for each pair the Qf1, the value of Qf if this pair is absent

pair     Qf1
   1   70.45
   2   75.61
   3   74.40
   4   73.81
   5   75.61
   6   72.09
   7   78.15
   8   74.40
   9   76.23
  10   73.81
  11   75.00
  12   78.15
  13   78.15
  14   80.17

and the characteristics of the movement scheme

Qf1av= 75.347   Qf1max=  80.17, for pair(s) 14
sd= 1.777, d4= 2.249, Qc=81.19, Qf=82.46, Qo=61.54

input file is 14p8rexpandedM.txt, output file is 14p8rScrEM.txt