令和元年度電気関係学会四国⽀部連合⼤会 講演論⽂集 (2019 新居浜⾼専)
2019 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (NIIHAMA)

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Improving Whale Optimization Algorithmby Using Logistic Map
Hidehiro SUGIOKA,Ken KAMIYOTSUMOTO, Yoko UWATE and Yoshifumi NISHIO( Tokushima University)

1. Introduction

ability of WOA. Logistic Map shows a chaotic behavior andis formulated by Eq. (6). We set the parameter α = 4.0 and
the initial value z(0) = 0.11.

Swarm intelligence algorithm is optimization algorithm
which simulate behavior of creature. Examples are Ant
Colony Optimization (ACO), Artificial Bee Colony Algorithm (ABC) and Whale Optimization Algorithm (WOA),etc.
In this study, we propose WOA improved by using Logistic Maps for random coeﬃcients. We compare results ofthe proposed method and the conventional WOA.

z(t + 1) = αz(t)(1 − z(t)).

4. Simulation result
We compare the proposed method to WOA with fortest functions. Formula, range and the optimal value ofeach function are shown in Table 1. x is variable in thefunction. xi denotes i-dimensional x.

2. Whale Optimization Algorithm

Table 1: Test functions

Whale Optimization Algorithm (WOA) is proposed by
Mirjalili and Lewis. WOA is nature-inspired meta-heuristic
optimization algorithm which mimics the feeding behaviorof humpback whales. There are three types of whale feeding
behavior: Search for prey, Encircling prey and Bubble-net
attacking. The procedure for WOA to search for answer isshown below.

Namef1{ −−−−→−−−−→ →−Drand = |Xrand − X (t)|→−−−−−→ →− −−−−→X (t + 1) = Xrand − A .Drand .

−(if |→A | > 1)

{ Formula[100(xi + 1 − x2i )2 + (xi − 1)2 ][-100,100]

0[-5.12,5.12]-12569[-5,5]-1.031630∑√−xi sin(i=1144x1 2 − 2.1x1 +3}|xi |)

x1 6 + x1 x2 − 4x2 2 + 4x2 4Each numerical experiment runs 20 times. In each test
function, we define tmax = 500, n = 30. Average value,
minimum value and maximum value are shown in Table 2.

Table 2: Results of WOA and proposed method
nameWOA

proposed method

f1

aveminmax1.144 × 10−94.648 × 10−14
5.572 × 10−96.584 × 10−4
3.397 × 10−13
2.961 × 10−3f2

aveminmax2.702 × 101
2.934 × 100
2.9714 × 101

1.150 × 101
4.482 × 10−2
2.9708 × 101

f3

aveminmax−1.230 × 104−1.2569 × 104−1.027 × 104−1.246 × 104−1.2569 × 104−1.171 × 104

f4

ave
min
max

−1.0307 × 100−1.0316 × 100−1.0281 × 100−1.0309 × 100−1.0316 × 100−1.0282 × 100

(1)
(2)

(3)

The proposed method obtains better value in cases of f2 ,f3 and f4 . However, minimum value is same value in f3 andf4 . WOA obtains better value in f1 .

{ −→−→→−′∗
D = |X (t) − X (t)|−(if |→A | ≤ 1, p > 0.5)−→′−→→−∗X (t + 1) = D el cos (2πl) + X (t)

30
∑

f4

Step 5. Repeat steps 2 to 4 and output the solution.
−(if |→A | ≤ 1, p ≤ 0.5)

Optimal value0f3

Step 4. Update the position of the whale by Eq. (3),−→Eq. (4) and Eq. (5). Equations are selected by | A |and p.

→−→− −→→−∗D = C .X (t) − X (t)|−→→−→− →−∗X (t + 1) = X (t) − A . D

Range[-100,100]i=1

Step 2. Calculate the fitness value fi and update the best−→
whale position X ∗ .
−→ −→Step 3. Update ⃗a, r⃗1 , r⃗2 , A , C , l and p. ⃗a is linearlydecreased from 2 to 0 over the course of iteration. r⃗1 ,r⃗2 and p are random coeﬃcient in [0,1]. l is random−→−→
coeﬃcient in [-1,1]. A and C is determined by Eq. (1)and Eq. (2).

{|xi |, 1 ≤ i ≤ 30max

f2

Step 0. Initialize the whales population n and the maximum number of iterations tmax .
−→Step 1. Set the position of the whale Xi . i is the numberof the whale.

−→A = 2⃗a.r⃗1 − ⃗a−→
C = 2r⃗2 .

(6)

(4)

5. Conclusion
We have proposed WOA with Logistic Map used for random coeﬃcients. We compared WOA and the proposed
method in four functions. Simulation results show that proposed method is better than WOA in three functions.
As our future works, We try to improve search performance by using other chaos for random coeﬃcient.

(5)

3. Proposed method
WOA has r⃗1 , r⃗2 , and l which are random coeﬃcients. We
insert Logistic Map to random coeﬃcients to improve search17