IEEE Computational Intelligence Magazine - November 2020 - 67

be updated each time. For either of the two types of updates
above, we use P j (x j, xlj) to denote the updated program, where
x j denotes the original drug in the prescription, and xlj
denotes the alternative drug in K x j . Therefore, for the set of
original prevention programs {P1, P2, f, PN }, each solution to
the problem can be represented by a (2N)-dimensional integer
vector x = {x 1, xl1, x 2, xl2, f, x N , xlN }, which indicates that the
x j -th drug in P j is to be replaced by its xlj -th alternative
^1 # j # N h; without loss of generality, x j = xlj denotes that
P j is unchanged.
Based on the efficacy of the original and alternative drugs,
we can determine the quantity of an alternative drug kl used
to replace an original drug k in the prescription. Based on the
change of the prescription, we can then determine the changes
of other medical resources, such as the types and quantities of
material and the working hours for processing the drugs. Consequently, we obtain the following attributes of the updated
prevention program P j (x j, xlj):
❏❏ The set U j (x j, xlj) of drugs; for each drug k ! U j (x j, xlj),
the quantity used per prescription is q Djk (x j, xlj);
❏❏ The set W j (x j, xlj) of other sharable resources used by the
program; for each resource k ! W j (x j, xlj), the quantity used
per prescription is q Gjk (x j, xlj);
❏❏ The set X j (x j, xlj) of other non-sharable resources used by
the program; for each resource k ! X j (x j, xlj), the quantity
used per prescription is q Fjk (x j, xlj).
The objective of the problem is to maximize the overall
effects of the updated prevention programs, provided that the
resources used by the programs do not exceed the available
resources. The effect of each updated program P j (x j, xlj) is
evaluated based on its deviation from the original program
P j; the larger the deviation, the smaller the effect is, as we
should trust the ability of TCM experts who develop the
original program. The deviation of P j (x j, xlj) from P j is
assessed in two aspects:
❏❏ The importance of drug x j in the original P j, which is
measured by a weight w jx j; a larger priority indicates a larger deviation;
❏❏ The priority of drug xlj in the alternative set K x j; a higher
priority indicates a smaller deviation.
Here, we calculate the deviation as follows:
	

DP j (x j, xlj) = w jx j I (K x j, xlj)(11)

where I (K x j, xlj) is the index of xlj in K x j (without loss of
generality, we set DP j (x j, x j) = 0 h .
Moreover, we use a weight w j to denote the susceptibility
of residents covered by program P j to the infectious disease,
and use a weight w li to denote the importance of each community i (which is related to the openness and population
density of the community). The objective of the problem is
defined as:
	

min f (x) =

The constraints of the problem are the quantities of each
drug, other sharable resource, and other non-sharable resource
used by the programs cannot exceed available quantities:
N

	

1 # k # K (13)

/ n j q Gjk (x j, xlj) # qt Gk ,

1 # k # K 1 	(14)

j=1
N

	
	

/ n j q Djk (x j, xlj) # qt Dk ,

N

j=1

/ n ij q Fjk (x j, xlj) # qt Fjk,

1 # i # M; 1 # k # K 2 (15)

j=1

It should be noted that, in Eqs. (13)-(15), we uniformly use
the operator ∑ for notational simplicity; however, it may not
always necessarily be summation. Typically, for drugs and material, ∑ denotes summation; for other resources such as devices and
personnel, ∑ can be other corresponding aggregation operators.
For example, supposing that a decocting machine can process 50
doses of a prescription, the operator will add 1 per 50 doses, and
will also add 1 if the number of remaining doses is less than 50.
B. Optimization Algorithm

A TCM prescription can have dozens of ingredients, and a
drug can have dozens of alternative drugs. Therefore, when the
number N of prevention programs is relatively large, the solution space of the problem can be very large, for which exact
optimization algorithms are often inefficient.
We use a metaheuristic optimization algorithm, water wave
optimization (WWO) [33], to efficiently solve the problem.
The algorithm starts by initializing a population of NP solutions. To evaluate the fitness of each solution x, we employ
three penalty functions v D (x), v G (x), and v F (x) to calculate the
violations of constraints (13), (14), and (15) as follows:
	
	

v D (x) =
v G (x) =

/ max e 0, / n j n D (x j, xlj) - nt D o (16)
K

N

k=1

j=1

K1

N

jk

/ max e 0, / n j n Gjk (x j, xlj) - nt kG o (17)

k=1

	

v F (x) =

k

j=1

K2

/ / max e 0, / n ij n F (x j, xlj) - nt F o(18)
M

N

jk

i=1 k=1

Long Waves of
Low Energy

ik

j=1

Short Waves of
High Energy

Ocean Floor

N

/ / w j w li DPj (x j, xlj)	(12)

j=1 i!Hj

FIGURE 2 Wave lengths of high-fitness and low-fitness waves
(solutions).

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IEEE Computational Intelligence Magazine - November 2020

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