Optimization of the CPT system
4.5 Optimization Algorithm
Optimization is becoming a critical tool for solving various problems in a wide range of fields such as engineering, science, technology and industry. Most of the real-world problems can be formulated as mathematical equations with some constraints involved.
The constraints can be associated with resource, time, cost or any other parameters upon which the objective function is dependent. Thus, the primary goal of the optimization technique is to find the most suitable solutions without violating the system constraints.
4.5.1 Single objective optimization (SOO)
There are many single objective optimization algorithms are available in the literature.
The majority of the algorithms are inspired by nature, as their search criteria are based on natural phenomena. The most popular algorithms are the genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization (ACO), and grasshopper algorithm (GOA), artificial bee colony algorithm (ABC), etc [108–112]. In the CPT optimization, a hybrid algorithm proposed in [113] is adopted to find the most suitable solutions.
The hybrid algorithm, ACOR-GA is formulated combining the crossover and mutation operation of the genetic algorithm (GA) with the ant colony optimization extended to
continuous domain (ACOR). The flowchart of the hybrid algorithm is illustrated in Fig.
4.4 and the process of the algorithm is discussed in the following steps.
Figure 4.4: Flowchart of the ACOR-G Aalgorithm. Figure reproduced from [113]
Step 1:The algorithm initiates after declaring the required parameters of ACOR, crossover, mutation, initial population (pop), initial population size and number of iterations. Then, the initial population is generated randomly within the defined parameters.
Step 2:The solution weights (ω) and selection probabilities (p) are calculated for an individual run based on the equations in [114].
Step 3: The mean (µ) and standard deviation of the initial population set are calculated which along with previously calculated (ω) and (p) affect the generation of the new population set in that particular iteration.
Step 4: A solution kernel is generated based on roulette wheel selection, which governs the generation of a new population named asnewpop. The process takes place 100 times i.e. equal to the number of the initial population size and the optimizing parameters are updated with each iteration.
Step 5:In this step, the algorithm performs crossover and mutation on randomly selected populations from the initial population sets and creates two new population sets namedpopcandpopm.
Step 6:All the solution sets i.e.newpop,popc,popmand initial population then merged together to generate an updated population of larger dimension.
Step 7:The updated population of the previous step is sorted based on the obtained cost from the objective functions and only the best 100 solutions are stored for the next iteration.
Step 8:The optimization process will be repeated from Step 3 until the maximum number of iterations is reached.
Step 9:Upon reaching the maximum iteration number, the whole process will be terminated and the best possible solutions will be displayed.
4.5.2 Multi-objective Optimization (MOO)
Multi-objective optimization problems mainly impose the challenge of satisfying multiple conflicting objective functions. While using evolutionary algorithms to solve multi-objective optimization problems, a non-dominated sorting technique [115] plays a crucial role in identifying non-dominated solutions to form Pareto optimal fronts [115].
Crowding distance calculation is also important to sort the population in order to decide which solutions to consider for the next generation [115]. In this study, the hybrid algorithm ACOR-G Ahas been incorporated with the non-dominated sorting technique to enhance the diversity among the generated solution through better exploration of the search space. The hybrid ACOR-G A algorithm was proposed in [?], where the crossover and mutation operation of the Genetic Algorithm (G A) was merged with the optimization
process of Ant Colony Optimization extended to the continuous domain (ACOR). The flowchart of the non-dominated sorting based ACOR-G Aalgorithm named MOACOR- G A is depicted in figure 4.5 and the process of applying the algorithm in solving the multi-objective optimization problem is described in the following steps.
Figure 4.5: Flowchart of the non-dominated sorting based ACOR-G A
Step 1:First of all, the initial population is randomly generated after declaring the various system parameters and the population size (N). For this study, the population size was taken to be 100.
Step 2: In this step, the fitness of the initial population is calculated using the objective functions declared in equations (4.16) and (4.17).
Step 3:In this stage, the non-dominated sorting technique is used to sort the initial population based on the fitness value. After that the crowding distance and the ranking of the current population are determined.
Step 4:The current solution is considered as the initial population of theACOR-G A algorithm and a new solution set of size N are generated.
Step 5:The new solution set is merged with the initial population increasing the total population size to 2N.
Step 6: In this step, the updated population undergoes a similar sorting process based on non-domination and crowding distance operator.
Step 7:The population size of 2N is truncated in this step by selecting the first N number of solutions that are considered as the fittest up to the current iteration.
Step 8:The updated population of the previous step is again sorted using the non- dominated sorting technique. Then, the population is further sorted by calculating the crowding distance of the solutions and ranking them accordingly.
Step 9:The best solution up to the current iteration is stored. Before reaching the maximum number of iterations the optimization process will be repeated from Step 4.
Step 10:Upon reaching the maximum iteration number, the whole process will be terminated and the best possible solutions will be displayed in a Pareto optimal front.