Notebook

Example of FIS-Mamdami with 2 Inputs and 1 Output¶

The Tipping Problem¶

In this example we're going to translate a concenptual problem: The Basic Tipping Problem. Given a number between 0 and 10 that represents the quality of service at a restaurant (where 10 is excellent), what should the tip be?

Tipping Problem — Both Service and Food Factors

If service is poor or the food is rancid, THEN tip is cheap
If service is good, THEN tip is average
If service is excellent or food is delicious, THEN tip is generous

For this system we have two input/output universe variables: service and food/ tip

Anastasiadis Stavros, stavrosana@aegean.gr / fuzzyelements.com

P.S. Special thank to JDWarner . You can view his comment and the original solution at Scikit-fuzzy mailing list: http://groups.google.com/group/scikit-fuzzy

In [8]:

%pylab inline
import numpy as np
import skfuzzy as fuzz
pylab.rcParams['figure.figsize'] = (10.0, 5.0)

Populating the interactive namespace from numpy and matplotlib

#### Step 1: Fuzzify Inputs / Output Variables

In [9]:

########## INPUTS ########################
#Input Universe functions
service = np.arange(0,11,.1)
food    = np.arange(0,11,.1)
# Input Membership Functions
# Service
service_poor = fuzz.gaussmf(service ,0,1.5)
service_good = fuzz.gaussmf(service ,5,1.5)
service_excellent = fuzz.gaussmf(service ,10,1.5)
# Food
food_rancid = fuzz.trapmf(food , [0, 0, 1, 3])
food_delicious = fuzz.gaussmf(food ,10,1.5)

########## OUTPUT ########################
# Tip
# Output Variables Domain
tip = np.arange(0,30,.1)
# Output  Membership Function 
tip_ch  = fuzz.trimf(tip, [0, 5, 10])
tip_ave = fuzz.trimf(tip, [10, 15, 25])
tip_gen = fuzz.trimf(tip, [20, 25, 30])

Step 2: Apply Fuzzy inputs to membership functions¶

Note: fuzz.fuzz_or allows different universes to be combined and fuzz.interp_membership allows inputs arbitrary values on the input universe. Comments from google scifit-fuzzy group by JDWarner

In more practical view: Say food = 4, we need to define the corresponding membership values of each sub-Fuzzy categories of fuzzified input variables.

It's like to say, IF food = 4 THEN has mf = 0.4 in Rancid Fuzzy Set and mf = 0 in Delicious Fuzzy Set

In [20]:

def food_category(food_in = 2):
    food_cat_rancid = fuzz.interp_membership(food,food_rancid,food_in) # Depends from Step 1
    food_cat_delicious = fuzz.interp_membership(food,food_delicious,food_in) # Depends form Step 1
    return dict(rancid = food_cat_rancid,delicious = food_cat_delicious)

def service_category(service_in = 4):
    service_cat_poor = fuzz.interp_membership(service,service_poor, service_in) # Depends from Step 1
    service_cat_good = fuzz.interp_membership(service,service_poor, service_in)
    service_cat_excellent = fuzz.interp_membership(service,service_excellent, service_in)
    return dict(poor = service_cat_poor, good = service_cat_good, excellent = service_cat_poor)

#Exaple input variables 
food_in = food_category(2.34)
service_in = service_category(1.03)
print "For Service", service_in
print "For Food ",food_in 

For Service {'poor': 0.62403967843338304, 'good': 0.62403967843338304, 'excellent': 0.62403967843338304}
For Food  {'delicious': 4.9972649113988357e-12, 'rancid': 0.33000000000000007}

Step 3: Determine the weight for each rule from fuzzy antecents -¶

IF part

In [16]:

rule1 = np.fmax(service_in['poor'],food_in['rancid'])
rule2 = service_in['good']
rule3 = np.fmax(food_in['delicious'],service_in['excellent'])

Step 4: Apply implication opetator (Mamdami - min):¶

Then part

In [17]:

imp1 = np.fmin(rule1,tip_ch)
imp2 = np.fmin(rule2,tip_ave)
imp3 = np.fmin(rule3,tip_gen)

Step 5: Aggregate all output - max¶

In [18]:

aggregate_membership = np.fmax(imp1, np.fmax(imp2,imp3))

Step 6: Defuzzify using Centroid¶

In [19]:

result_tip = fuzz.defuzz(tip, aggregate_membership , 'centroid')
print result_tip

15.4100253375