A New Improved Quantum Evolution Algorithm with Local Search Procedure for Capacitated Vehicle Routing Problem

Table 2

Different quantum rotation angles of QEA.

No.	Different quantum rotation angles	Authors
1	Type I: Δ𝜃^𝑡𝑗,𝑖=sgn(𝛼^𝑡𝑗,𝑖𝛽^𝑡𝑗,𝑖)𝜃0exp(|𝛽^𝑡𝑗,𝑖|/|𝛼^𝑡𝑗,𝑖|)+𝛾), where 𝜃0 is an initial rotation angle and 𝛾 is a scale parameter; type II: 𝛼^𝑡+1𝑗,𝑖=𝛼^𝑡+1𝑗,𝑖/(fix(𝑘/5)+1);𝛽^𝑡+1𝑗,𝑖=1−(𝛼^𝑡+1𝑗,𝑖)^2, where fix(⋅) is a round function and 𝑘 denotes the number of continuously invalid evolution alleles	Zhang and Gao [20]
2	𝜃=−sgn(𝐴)×𝜃0×(‖∇fixmax−∇fixmin‖/‖∇fix(𝑋)−∇fixmin‖), where 𝐴 is sin(𝜃1−𝜃0), 𝜃0 and 𝜃1 are the initial and the current best rotation angle, ‖⋅‖ is a certain norm, and ∇fix(⋅) is the gradient function of fitness at 𝑋.	P. L. Li and S. Y. Li [21]
3	𝜃𝑖=𝑥Δ𝜃𝑖(1+𝑥𝜆𝑖); Δ𝜃𝑖=∑^𝐾𝑛=1(𝜔^𝑛𝑏𝑖−𝜔𝑖)/𝐾; 𝑥=sgn(𝛼𝑖𝛽𝑖(𝑏𝑖−0.5)); 𝜆0=𝑓(𝑋𝑖); 𝜆𝑡+1=𝜇𝜆𝑡(1−𝜆𝑡), where 𝜔𝑖 represents the 𝑖th rotation angle and 𝜔𝑏𝑖^𝑛 the 𝑛th best individual's 𝑖th rotation angle; the last two equations are the logistic equations	Hu and Wu [22]
4	𝜃𝑖=𝜆⋅Φ^𝑗𝑘⋅Φ^𝑗𝑘; Φ^𝑗𝑘=8Φ^𝑗𝑘−1(1−Φ^𝑗𝑘−1)−1; 𝜆=𝛾𝜆0((1−𝐾)/𝐾), where the second equation is a modified chaos logistic equation, 𝜆0 is a control parameter, 𝛾 is a temporary parameter, and generally 𝛾=1.	Gao and Wang [23]
5	Δ𝜃=minΔ𝜃+(maxΔ𝜃−minΔ𝜃)(𝜔1⋅(‖𝑥−𝑏‖/𝑛)+𝜔2⋅|𝑓(𝑥)−𝑓(𝑏)|/max{𝑓(𝑥),𝑓(𝑏)}), where maxΔ𝜃 and minΔ𝜃 represent the maximum and minimum rotation angle separately, 𝜔1 and 𝜔2 are two predetermined constant numbers, and 𝑛 is the length of its binary chromosome.	Chen et al. [24]
6	𝜃=𝑘⋅𝑓(𝛼𝑖,𝛽𝑖); 𝑘=𝜋⋅exp(−𝑡/itermax), where 𝑡 is the current iteration of the algorithm and itermax is the max iteration of the algorithm.	Lee et al. [25]