ZMF

Generates a Z-function fuzzy membership generator, named such because of its Z-like shape.

The function begins falling from 1 at the ‘ceiling’ and reattains zero at the ‘foot’.

Excel Usage

=ZMF(x, a, b)

• x (list[list], required): Array of independent universe variables.
• a (float, required): Ceiling, where the function begins falling from 1.
• b (float, required): Foot, where the function reattains zero.

Returns (list[list]): Array of membership values corresponding to the input universe.

Example 1: Z-function membership over a positive domain

Inputs:

x						a	b
0	1	2	3	4	5	1	4

Excel formula:

=ZMF({0,1,2,3,4,5}, 1, 4)

Expected output:

Result
1
1
0.777778
0.222222
0
0

Example 2: Z-function membership before the descent starts

Inputs:

x	a	b
0	1	4

Excel formula:

=ZMF(0, 1, 4)

Expected output:

1

Example 3: Z-function membership with a wide transition band

Inputs:

x						a	b
-2	-1	0	1	2	3	-1	2

Excel formula:

=ZMF({-2,-1,0,1,2,3}, -1, 2)

Expected output:

Result
1
1
0.777778
0.222222
0
0

Example 4: Z-function membership on a negative domain

Inputs:

x					a	b
-5	-4	-3	-2	-1	-4	-2

Excel formula:

=ZMF({-5,-4,-3,-2,-1}, -4, -2)

Expected output:

Result
1
1
0.5
0
0

Python Code

Show Code

import numpy as np
from skfuzzy import zmf as fuzz_zmf

def zmf(x, a, b):
    """
    Generate a Z-function fuzzy membership generator.

    See: https://pythonhosted.org/scikit-fuzzy/api/skfuzzy.html#skfuzzy.zmf

    This example function is provided as-is without any representation of accuracy.

    Args:
        x (list[list]): Array of independent universe variables.
        a (float): Ceiling, where the function begins falling from 1.
        b (float): Foot, where the function reattains zero.

    Returns:
        list[list]: Array of membership values corresponding to the input universe.
    """
    try:
        def to1d(arr):
            if isinstance(arr, list):
                flat = []
                for row in arr:
                    row_list = row if isinstance(row, list) else [row]
                    for val in row_list:
                        try:
                            flat.append(float(val))
                        except (TypeError, ValueError):
                            continue
                return np.array(flat)
            return np.array([float(arr)])

        x_arr = to1d(x)
        if len(x_arr) == 0:
            return "Error: Input mapping array x cannot be empty"

        if a > b:
            return "Error: Parameters must satisfy a <= b"

        result = fuzz_zmf(x_arr, a, b)
        return [[float(val)] for val in result]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

x *

Array of independent universe variables.

a *

Ceiling, where the function begins falling from 1.

b *

Foot, where the function reattains zero.