5. Hybrid NSGA II and MOPSO algorithm for solving CEED problem

The mechanism of the proposed hybrid approach for solving the CEED problem is to integrate the desirable features of NSGA II (retaining the elitism feature) and MOPSO (exploitation capability) while curbing the individual flaws (NSGAII––does not have an efficient feedback mechanism, PSO overutilization of resources). The mechanism to explore the search space differs in both the algorithms. GA uses mutation and crossover operators which will enhance the exploration task of the hybrid algorithm. The particles in PSO are influenced by their own knowledge and information shared among swarm members. PSO enhances the exploitation task of the hybrid algorithm by finding better solutions from the good ones by searching the neighborhood of good solutions. In this hybrid algorithm at every generation, the Pareto dominance of the population is computed and based on these values non dominated sorting is performed [19]. In order to avoid premature convergence, the elite upper half of the population are enhanced by NSGA II algorithm while the lower

half of the population are considered as swarm particles and are optimized by MOPSO to make them converge around the best possible solutions. The hybrid NSGA II-MOPSO algorithm for solving the CEED problem is stated below:

• Perform Constraint Handling Mechanism

• Merge the parent and offspring population

• Each front is filled in ascending order

• Store the non-dominated solutions in list Ϝ<sup>1</sup>

• Plot the non-dominated solutions in list Ϝ<sup>1</sup>

• Initialize external repository rep • Create grid and find grid index

• Select leader from external repository • Update particle position and velocity

• Apply Mutation and calculate new solutions

• Apply Constraint Handling Mechanism

• Add non dominated particles to the repository

• Determine domination of new repository members

• Keep only the non-dominated members in the repository

• Constraint Handling Mechanism

• Determine Domination pBest

• Update grid and grid index

• Modify inertia weight

• Last front- descending order of crowding distance

• Store the values of the particles as their personal best pBest

• Perform non domination sorting

• Select solutions

• Determine Domination

• For each particle do

• End for

• Evaluate the fuel cost objective function E and emission objective function F

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• Calculate crowding distance and rank based on non-domination fronts

• Position and cost of the particle are initialized from the lower half of the population

• Evaluate the fuel cost objective function E and emission objective function F

	- Population Size nPop
	- Maximum number of iteration MaxIt
	- Crossover Percentage pCrossover
	- Mutation Percentage pMutation
	- Mutation rate mu
	- Mutation step size sigma
	- Repository size nRep
	- Inertia Weight w and Inertia Weight damping rate wdamp
	- Personal learning coefficient c1 and Global learning coefficient c2
	- Number of grids per dimension nGrid
	- Inflation Rate alpha, leader selection pressure beta, Deletion selection pressure gamma
	- Mutation rate mu
	- Generate a random nPop size population
	- Once the random population is initialized the Constraint Handling Mechanism proposed in Section 3 is carried out.
	- Evaluate the fuel cost objective function E and emission objective function F
	- Perform Non Domination Sorting
	- Calculate Crowding Distance and rank the population based on Non Dominated fronts
	- Truncate and divide the population into two halves.
	- Using the upper half of the population create offspring population
		- Selection, Crossover and Mutation

half of the population are considered as swarm particles and are optimized by MOPSO to make them converge around the best possible solutions. The hybrid NSGA II-MOPSO

algorithm for solving the CEED problem is stated below:

• Specify the parameters for the CEED problem • Specify the parameters for NSGA II Algorithm

• Maximum number of iteration MaxIt

• Specify the parameters for MOPSO Algorithm

• Number of grids per dimension nGrid

• Generate a random nPop size population

posed in Section 3 is carried out.

• Perform Non Domination Sorting

• Truncate and divide the population into two halves.

• Selection, Crossover and Mutation

• Using the upper half of the population create offspring population

• Inertia Weight w and Inertia Weight damping rate wdamp

• Personal learning coefficient c1 and Global learning coefficient c2

• Inflation Rate alpha, leader selection pressure beta, Deletion selection pressure gamma

• Once the random population is initialized the Constraint Handling Mechanism pro-

• Calculate Crowding Distance and rank the population based on Non Dominated

• Evaluate the fuel cost objective function E and emission objective function F

• Crossover Percentage pCrossover • Mutation Percentage pMutation

• Population Size nPop

92 Particle Swarm Optimization with Applications

• Mutation rate mu

• Mutation step size sigma

• Repository size nRep

• Mutation rate mu

• Evaluate the objective functions

• Initialize Population

• For each generation do

fronts

	- Initialize external repository rep
	- Create grid and find grid index
	- Select leader from external repository
	- Update particle position and velocity
	- Constraint Handling Mechanism
	- Evaluate the fuel cost objective function E and emission objective function F
	- Apply Mutation and calculate new solutions
	- Apply Constraint Handling Mechanism
	- Determine Domination pBest
