## 2 Truck Platooning Speed Control Based on Multi-Objective Optimization
### Optimal Speed Computation Based on Multi-Objective Optimization
#### 2.2.2 Multi-Objective Speed Optimization  
This paper proposes objective functions for bottleneck cell traffic flow, overall vehicle travel time, and truck platooning fuel consumption, where the former must be maximized, whereas the latter two must be minimized. To optimize the truck platooning control speed, each objective function can be weighted to convert the multi-objective optimization problem into a single-objective optimization problem. However, it is unclear how to reasonably determine the weights of specific objectives, or approach the magnitude between each objective. Instead, the non-dominated ranking genetic algorithm with elite strategy (NSGA-II) was employed to solve multi-objective optimization in this study.  
Genetic algorithms, also known as simple genetic algorithms (SGAs), are on the basis of Darwinian evolutionary theory. An SGA is a type of stochastic global search optimization algorithm that takes all individuals in a population as the object, and starts from any initial population. Through random selection, crossover, and mutation operations, the algorithm generates a population group more adapted to the environment, and gradually evolvesthe population to a better region in the search space. Finally, it converges to a group of individuals that are most adapted to the environment through continuous reproduction and evolution, thus obtaining an optimal solution. The basic processes of genetic algorithms include chromosome encoding, generation of the initial population, calculation of the fitness value, genetic operator operation, and genetic termination.  
An SGA can be represented as:  
\[SGA=(C,\ E,\ P_{0},\ M,\ \phi,\ \Gamma,\ \psi,\ T) \tag{13}\]  
where, \(C\) denotes the coding method for individuals, \(E\) denotes the individual fitness evaluation function, \(P_{0}\) denotes the initial population, \(M\) denotes the population size, \(\phi\) denotes the selection operator, \(\Gamma\) denotes the crossover operator, \(\psi\) denotes the variation operator, and \(T\) denotes the genetic termination condition.  
Non-dominated sorting genetic algorithms (NSGAs) are an extension of SGAs using the concept of Pareto optimality. Typically, these algorithms yield an optimal solution set that is impossible to optimize further. Such a solution set is called a non-dominated or Pareto optimal solution set.  
Unlike SGAs, NSGAs first compute and rank the dominant and non-dominant relationships among all individuals in a population, and then perform the genetic operation. The Pareto optimal solution set provides a non-inferior solution to the multi-objective optimization problem, and the decision maker must select a value from the set as the optimal solution through a method known as problem-specific analysis. In this study, the speed corresponding to the lowest overall vehicle travel time was selected as the optimal solution from the perspective of traffic efficiency.  
Because NSGA algorithms typically entail high computational complexity and the loss of the best individual, this study adopted the non-dominated sorting genetic algorithm with elite strategy (NSGA-II). NSGA-II presents the following advantages: (1) a fast non-dominated sorting method, which reduces computational complexity; (2) a crowding degree and crowding degree comparison operator, which maintains population diversity; and (3) an elite strategy to expand the sampling space. In this study, key parameters of the NSGA-II are a population size of 100, and 200 termination iterations.