PIMF

Pi-function fuzzy membership generator. Equivalently a product of the S-function (smf) and Z-function (zmf).

The function begins to climb from zero at the left ‘foot’ (a), levels off at 1 at the left ‘ceiling’ (b), begins falling from 1 at the right ‘ceiling’ (c), and reattains zero at the right ‘foot’ (d).

Excel Usage

=PIMF(x, a, b, c, d)

x (list[list], required): Array of independent universe variables.
a (float, required): Left foot, where the function begins to climb from zero.
b (float, required): Left ceiling, where the function levels off at 1.
c (float, required): Right ceiling, where the function begins falling from 1.
d (float, required): Right foot, where the function reattains zero.

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

Example 1: Pi-function membership with a flat top

Inputs:

x						a	b	c	d
0	1	2	3	4	5	1	2	3	4

Excel formula:

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

Expected output:

Result
0
0
1
1
0
0

Example 2: Pi-function membership inside the plateau

Inputs:

x	a	b	c	d
2.5	1	2	3	4

Excel formula:

=PIMF(2.5, 1, 2, 3, 4)

Expected output:

1

Example 3: Pi-function membership with wider transition regions

Inputs:

x							a	b	c	d
0	1	2	3	4	5	6	1	2.5	3.5	5

Excel formula:

=PIMF({0,1,2,3,4,5,6}, 1, 2.5, 3.5, 5)

Expected output:

Result
0
0
0.777778
1
0.777778
0
0

Example 4: Pi-function membership over an asymmetric support

Inputs:

x						a	b	c	d
-2	-1	0	1	2	3	-1	0	1.5	3

Excel formula:

=PIMF({-2,-1,0,1,2,3}, -1, 0, 1.5, 3)

Expected output:

Result
0
0
1
1
0.777778
0

Python Code

Show Code

import numpy as np
from skfuzzy import pimf as fuzz_pimf

def pimf(x, a, b, c, d):
    """
    Generate a Pi-function fuzzy membership generator.

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

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

    Args:
        x (list[list]): Array of independent universe variables.
        a (float): Left foot, where the function begins to climb from zero.
        b (float): Left ceiling, where the function levels off at 1.
        c (float): Right ceiling, where the function begins falling from 1.
        d (float): Right 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 not (a <= b <= c <= d):
            return "Error: Parameters must satisfy a <= b <= c <= d"

        result = fuzz_pimf(x_arr, a, b, c, d)
        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 *

Left foot, where the function begins to climb from zero.

b *

Left ceiling, where the function levels off at 1.

c *

Right ceiling, where the function begins falling from 1.

d *

Right foot, where the function reattains zero.