Journal of Applied Mathematics

Research Article

An Improved Particle Swarm Optimization for Solving Bilevel Multiobjective Programming Problem

Table 1

The notations of the algorithm.
π‘₯ 𝑖 The 𝑖 t h particle’s position of the upper level problem.
𝑣 π‘₯ 𝑖 The velocity of π‘₯ 𝑖 .
𝑦 𝑗 The 𝑗 t h particle’s position of the lower level problem.
𝑣 𝑦 𝑗 The velocity of 𝑦 𝑗 .
𝑧 𝑗 The 𝑗 t h particle’s position of BLMPP.
𝑝 𝑝 b e s t 𝑦 𝑗 The 𝑗 t h particle’s personal best position for the lower level problem.
𝑝 𝑝 b e s t π‘₯ 𝑖 The 𝑖 t h particle’s personal best position for the upper level problem.
𝑝 𝑔 b e s t 𝑙 The particle’s global best position for the lower level problem.
𝑝 𝑒 𝑔 𝑏 𝑒 𝑠 𝑑 The particle’s global best position for the upper level problem.
𝑁 𝑒 The population size of the upper level problem.
𝑁 𝑙 The subswarm size of the lower level problem.
𝑑 Current iteration number for the overall problem.
𝑇 The predefined max iteration number for 𝑑 .
𝑑 𝑒 Current iteration number for the upper level problem.
𝑑 𝑙 Current iteration number for the lower level problem.
𝑇 𝑒 The predefined max iteration number for 𝑑 𝑒 .
𝑇 𝑙 The predefined max iteration number for 𝑑 𝑙 .
𝑀 𝑒 Inertia weights for the upper level problem.
𝑀 𝑙 Inertia weights the lower level problem.
𝑐 1 𝑒 The cognitive learning rate for the upper level problem.
𝑐 2 𝑒 The social learning rate for the upper level problem.
𝑐 1 𝑙 The cognitive learning rate for the lower level problem.
𝑐 2 𝑙 The social learning rate for the lower level problem.
N D 𝑒 Nondomination sorting rank of the upper level problem.
C D 𝑒 Crowding distance value of the upper level problem.
N D 𝑙 Nondomination sorting rank of the lower level problem.
C D 𝑙 Crowding distance value of the lower level problem.
𝑃 𝑑 The 𝑑 t h iteration population.
𝑄 𝑑 The offspring of 𝑃 𝑑 .
𝑆 𝑑 Intermediate population.