Which model can I use for this problem with multiple inputs and outputs? Which model is the most appropriate for this problem with multiple inputs and outputs?

The data set is

A1, A2, A3, A4, A5, A6, B1, B2, B3, B4


where A1, A2, A3, A4, A5, A6 are the inputs and B1, B2, B3, B4 the outputs (this is what I want the model to predict).

What an LSTM be appropriate for this task? Any advice or hint would be much appreciated. Also if anyone can share already done examples, it would really help me a lot.
This depend on type of data you use. 

Time sequence data

If the data advanced in time, a LSTM or similar RNN should be used. RNN calculate output through time. It works very good on time series data as it have a real sense of time. While CNN and MLP could work for time series data, it often don't work that well as different timestep of data is not defined. 

Non- Time sequence data

According to you previous comment, the data seems to be of this kind. In the case of your data, a normal Multi-layer perceptron works well for this. The data is a direct mapping between the input and the output. If the input data is an image, use a CNN. 

For example code in keras, see here:
You need the pandas module for this to work. Run pip install pandas to install pandas.
    from keras.models import Sequential
    from keras.utils import np_utils
    from keras.layers.core import Dense, Activation, Dropout

import pandas as pd
import numpy as np

# Read data
train = pd.read_csv('../input/train.csv')
labels = train.ix[:,0].values.astype('int32')
X_train = (train.ix[:,1:].values).astype('float32')
X_test = (pd.read_csv('../input/test.csv').values).astype('float32')

# convert list of labels to binary class matrix
y_train = np_utils.to_categorical(labels) 

# pre-processing: divide by max and substract mean
scale = np.max(X_train)
X_train /= scale
X_test /= scale

mean = np.std(X_train)
X_train -= mean
X_test -= mean

input_dim = X_train.shape[1]
nb_classes = y_train.shape[1]

# Here's a Deep Dumb MLP (DDMLP)
model = Sequential()
model.add(Dense(128, input_dim=input_dim))
model.add(Activation('relu'))
model.add(Dropout(0.15))
model.add(Dense(128))
model.add(Activation('relu'))
model.add(Dropout(0.15))
model.add(Dense(nb_classes))
model.add(Activation('softmax'))

# we'll use categorical xent for the loss, and RMSprop as the optimizer
model.compile(loss='categorical_crossentropy', optimizer='rmsprop')

print("Training...")
model.fit(X_train, y_train, nb_epoch=10, batch_size=16, validation_split=0.1, show_accuracy=True, verbose=2)

print("Generating test predictions...")
preds = model.predict_classes(X_test, verbose=0)

def write_preds(preds, fname):
    pd.DataFrame({"ImageId": list(range(1,len(preds)+1)), "Label": preds}).to_csv(fname, index=False, header=True)

write_preds(preds, "keras-mlp.csv")


Code source: https://www.kaggle.com/fchollet/simple-deep-mlp-with-keras

In conclusion, in the case of our data, use a multi layer perceptron  should work. Hope kit can help you and have a nice day!