Page 109 - ITUJournal Future and evolving technologies Volume 2 (2021), Issue 1

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 1

Algorithm 1 Lightweight normalization
Require: the raw value of each attribute for each

candidate
for each attribute do

if is an upward attribute then

+
is the upper bound of

= + Fig. 2 – FiPy board from Pycom [18].

else if is a downward attribute then

−
is the lower bound of

−
=

end if
end for
return the normalized value of

−
+
= { , | = 1, 2, ..., } is composed of the up‐

per and lower bounds of each attribute , such that ∀ ∈
, 0 < < +∞. B is stable, thus normalized values from
the alternatives will not be altered by the removing of any
other alternative. This completely eliminates rank rever‐
Fig. 3 – Pytrack sensor shield from Pycom [18].
sal and reduces algorithmic complexity at once. Indeed,
Equation (1) requires the computation of the denomina‐
2
tor √∑ for each value of (for attributes). This
=1 √
√
+
is not required with our bounded normalization and only = √ ∑(1 − ) 2

the division between the bound and the value is com‐ ⎷ =1 (10)
puted. Knowing the ixed bounds allows us to simplify √
√
2
−
TOPSIS further: Equation (3) is used to establish the ideal = √ ∑

positive and negative alternatives. Extreme values are ⎷ =1
found according to Equation (4) for upward attributes or
Equation (5) for downward ones. Those operations re‐ 5.2 Complexity
quire many comparisons. With bounded normalization, The reduced complexity of our algorithm can be assessed
we can simplify the determination of the ideal alterna‐ with an algorithmic complexity comparison. As big
−
+
tives: determination of and is trivial, as the nor‐ notation is only pertinent for large inputs, we choose
malized maximum and minimum bounds of the attributes to quantify the number of operations spared with our
are respectively equal to 1 and 0. Thus, Equation (4)
and Equation (5) can be simpli ied by Equation (9). In method instead of classic TOPSIS. We consider one oper‐
turn, determination of the ideal alternatives according ation as one of the four basic arithmetic operations: ad‐
Equation (3) shows that these are static and shown in dition, subtraction, multiplication and division. We also
Equation (8). Finally, distances computation according consider square root and value comparison as a single op‐
to Equation (6) can be simpli ied by Equation (10). In‐ eration. This is just an estimation and is not exact as a
deed, as the ideal alternatives are known and static, we square root is decomposed into multiple simpler opera‐
+
−
thus know that = 1 and = 0. tions when computed. However, as the exact decomposi‐

tion is dependent on the hardware, it is irrelevant to as‐
Those simpli ications reduce the complexity of the TOP‐ sign a precise operation cost to a square root. Hereafter
SIS method. Moreover, as the normalization uses a stable we consider and to be the dimensions of the decision
referential, rank reversal probability is eliminated. Those matrix.
modi ications thus reduce the time required for execu‐ Firstly, Equation (1) requires at least 3 operations,
tion, as we will see in Section 10. while using Algorithm 1 instead reduces it to oper‐
ations. Replacing equations (3), (4) and (5) by Equa‐
tions (8) and (9) spares the cost of the min‐max algorithm,
+
= [1...1] thus 2( − 1) operations. Finally, using Equation (10)
(8)
−
= [0...0] instead of Equation (6) spares operations. Our propo‐
sition thus spares a total of 5 − 2 operations.
6. SELECTION EXPERIMENTS & RESULTS
+
= 1 We implemented both algorithms in MicroPython on FiPy

−
= 0 (9) modules from Pycom, coupled with Pytrack expansion

© International Telecommunication Union, 2021 93