Multiclass Multioutput machine learning or neural network problem in python

Hello everyone,

My data has
5 input features(columns) ( M1 , M2, M3, M4, M5) which are float numbers.
6 output labels(columns) (E1, E2, E3 ,E4 ,E5 ,E6) , which contain discrete numbers (classes) 0 to 8 such as 0,1,2,3,4,5,6,7,8. , for example samples look likes this :

M1     M2    M3     M4     M5          E1      E2      E3      E4     E5      E6

7.6   15.2   38.7  54.8   67.5         0        0       2       1       0       0
7.8   16.5   39.7  64.6   77.5         8        0       0       0       5       0
8.8   26.5   49.7  74.8   87.5         0        7       0       0       0       6
9.8   28.5   50.7  76.8   89.5         1        0       3       0       0       0

Now,this looks like to me,multiclass multioutput classification , what is best solution to this problem. Solution which gives highest acuuracy or best prediction.
Should i use, machine learning classification approach or nural network approach.
I have tried few of methods, but could not get desired reults.
csv file attached , which is not too big.
Data.csv (11.9 KB)
Need help in this regard
Thanks.

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