<AddIn language="PYTHON" library="PythonAddIn_addin.py" namespace="PythonAddIn_addin">
from Install.businesslogic import Analytics
Rather than Install.businesslogic, does import businesslogic work? It should. If not, try sys.path.append(os.path.dirname(__file__)) before your import thought I thought I made relative imports work.
# Import os, sys; append relative directory to path import os import sys sys.path.append(os.path.dirname(__file__)) import arcpy import pythonaddins # Import your modules and add functions to namespace from businesslogic import *