1. Non-Linear Problems General approach

2. Non-linear Optimization Many objective functions, tend to be non-linear. Design problems for which the objective is to minimize cost or maximize benefits minus costs usually have cost functions with economies of scale. This implies a non-linear function

3. Non-linear Optimization Various approaches exist for solving non- linear problems. One of these is to divide the nonlinear functions into several linear sections (piecewise linearization). Another approach would be Genetic Algorithms

4. GA It is robust and computationally efficient for many types of problems, especially those that are highly nonlinear It is based on Theory of Evolution

5. GA Steps Step 1: Population Generation: A population of n chromosomes (i.e., individuals) is generated by randomly selecting values for the genes in the chromosomes. (I.e., randomly assign values to the decision variables for each of a large number of alternatives.) Step 2: Fitness Evaluation: Evaluate the “fitness” of each chromosome in the population. (I.e., calculate the value of the objective function for each alternative.) Step 3: Test for Completion: Test to see if an end condition has been achieved (e.g., test to see if a maximum number of generations has been reached, etc.). If so, stop. If not, continue with the next step.

6. Step 4: Create a New Population: Apply the processes of selection, crossover, mutation, and replacement to build a new population. – Step 4a: Selection: Select two parent chromosomes from the present population according to their fitness: the greater the fitness of an individual, the greater is the chance that the individual will be selected to be a parent and produce offspring. (I.e., select two alternatives from the current collection of alternatives, and base that selection upon the value of the objective function of the current alternatives.) – Step 4b: Crossover: With a pre-selected probability, select genes from one parent or the other to form a new individual (i.e., to form an offspring). (I.e., use some of the decision variable values from one of the alternatives, and some from the other, to formulate a new alternative.) – Step 4c: Mutation: With a pre-selected probability, cause a mutation to happen at any given gene in the new individual (i.e., make a small change in the value of a randomly selected decision variable). (I.e., make small, random changes in the values of some of the decision variables of the new alternative.)

7. Selection Process

8. Crossover Process

9. Mutation Process Example