Operators

Encoding

The encoding mechanism defines how solutions are represented and manipulated in the optimization process. In iMOPSE, solutions are encoded in a structured format that is suitable for various problem domains and operators.

The encoding structure consists of multiple sections, each defined by a specific type of encoding and a range of values for each gene. The types of encoding used can be:

• Float: Represents for example relationships between different components in MSRCPSP.
• Integer (Permutation): Represents sequences or orderings, useful for problems like the Traveling Salesman Problem.
• Binary: Uses binary values (0s and 1s) to represent the presence or absence of certain features.

Each section of the encoding describes the minimum and maximum values for each gene, ensuring that the generated solutions adhere to the problem's constraints.

The encoding structure is crucial for the operators used in the optimization process:

• Crossover Operators: Combine the genetic material of parent solutions according to the encoding scheme, ensuring that the offspring inherit the properties of their parents appropriately.
• Mutation Operators: Introduce random changes to the genes in the encoding, promoting diversity and helping to explore the search space more effectively.

Overall, the encoding mechanism in iMOPSE provides a flexible and powerful way to represent solutions, facilitating the application of various operators to optimize complex problems effectively.

Crossover Operators

Crossover operators combine two parent solutions to generate new offspring. Here are some of the crossover operators used in iMOPSE:

• Binomial Crossover: Binomial Crossover creates offspring by combining the genetic material of two parents based on a binomial distribution. Each gene in the offspring is taken from one of the parents with a probability defined by the crossover rate. This method ensures a good mix of genetic material, promoting diversity in the population.
• Uniform Crossover: In Uniform Crossover, each gene in the offspring is selected randomly from either of the parent genes with equal probability. This method maintains genetic diversity by allowing a uniform distribution of genes from both parents.
• Order Crossover (OX): Order Crossover is used for problems where the order of genes is important, such as the Traveling Salesman Problem. It creates offspring by copying a segment of one parent’s chromosome and filling the remaining genes from the other parent while preserving the order of genes. This method ensures that the offspring inherit the relative order of genes from both parents.
• Cycle Crossover (CX): Cycle Crossover ensures that each gene in the offspring comes from one of the parents by maintaining the cycle structure. It identifies cycles between the two parents and exchanges the corresponding genes to create offspring. This method preserves the relative ordering of genes, which is crucial for permutation-based problems.
• Position-Based Crossover: Position-Based Crossover selects a subset of positions in the offspring and fills them with genes from one parent, while the remaining positions are filled with genes from the other parent. This method maintains the positional information of genes, which can be important for certain types of optimization problems.

Mutation Operators

Mutation operators introduce random changes to individual solutions. Here are some of the mutation operators used in iMOPSE:

• Cheapest Resource Mutation: Cheapest Resource Mutation modifies an individual by assigning the cheapest available resource to a task. This method helps in reducing the overall cost of the solution while maintaining feasibility.
• Reverse Flip Mutation: Reverse Flip Mutation modifies an individual by reversing the sequence of genes between two randomly selected positions. This method helps in exploring new solutions by changing the order of tasks or resources.
• Differential Evolution Mutation: Differential Evolution Mutation generates new candidate solutions by adding the weighted difference between two randomly selected individuals to a third individual. This method is effective in exploring the search space and finding global optima.
• Random Bit Mutation: Random Bit Mutation flips a random bit in the binary representation of an individual. This method introduces small random changes to the solution, helping to maintain genetic diversity and avoid premature convergence.
• Swap Mutation: Swap Mutation selects two positions in the individual and swaps their values. This method is particularly useful for permutation-based problems, as it maintains the validity of the permutation while exploring new solutions.