Plagiarism Declaration
2. Literature Review
2.5. Optimisation
As mentioned previously, this investigation deals with the expansion of the optimisation model proposed and demonstrated by Chang & van Zyl (2012) to include pumping systems.
This section will focus on a description and basic review of the optimisation method employed by Chang & van Zyl (2012). A more mechanistic approach, detailing the changes and additions made, will be given in the methodology.
2.5.1. Objectives of optimisation
Any optimisation technique has the goal of finding the most efficient solution within the specified constraints. As previously mentioned, the two most prominent factors, when designing a bulk water supply system, are the reliability of the system (represented as the inverse of failure frequency) and the total financial cost of the system. The pareto-optimal obejective is the minimisation of both cost and failure frequency.
The failure frequency tolerance proposed by van Zyl et al. (2008) of 1 failure in 10 years under seasonal peak conditions, suggests that using cost as the sole objective allows for the possibility that the system is cost efficient but not sufficiently reliable for the proposed development. This possibility demonstrates that multi-objective optimisation techniques are important in order to maximize the net benefit (in this case, in terms of reliability), defined as the benefit minus cost (Walski, 2001).
2.5.2. Genetic optimisation
When considering the various solutions for a given bulk water supply problem, the number of unique solutions available for a reasonably sized system are, by orders of magnitude, too large to realistically calculate the cost and reliability of each solution. This is especially true when considering the stochastic nature of determining reliability and the large associated computational load for a single solution. This necessitates an optimisation technique that is representative of the entire solution space and can sample and refine solutions without having to analyse a large number of unique solutions.
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As the name suggests, a genetic algorithm is an optimisation technique that is based on the mechanics of natural selection and genetics (Holland, 1975; Goldberg, 1989 as cited by Chang & van Zyl, 2012). It selects, combines and manipulates possible solutions in the same way that nature permits survival, reproduction and the combination of chromosomes in search of the best adaptation. (Murphy & Simpson, 1992, as cited by Chang & van Zyl, 2012)
Genetic Algorithm: NSGA-II
As the genetic algorithm employed in the model suggested by Chang & van Zyl (2012) is a non-dominated sorting genetic algorithm (NSGA), the scope of the review will be limited as such. The basic operation of the NSGA-II algorithm is as follows:
The population size, decision variables, decision variable boundaries, number of generations, objectives and other parameters specific to the optimisation of the algorithm are inputted. A population is initialized by generating randomly within the solution space defined and bounded by the decision variables' limits. The decision variables in this application are the system components (In this instance, pump power, number of pipes, pipe size, reservoir size etc.). The population consists of a set of chromosomes representing a certain assortment of genes (decision variables).
The population is evaluated in terms of fitness; each chromosome is evaluated in terms of the objectives set (cost and reliability, in this instance). The fitness of each unique solution (chromosome) is translated into a probability for selection, with fitter solutions having a higher probability of being selected and recombined into the solution population.
Each solution set is sorted in the non-domination sorting algorithm. The algorithm is based on the NSGA definition of domination and involves ranking, sorting into domination fronts and assigning a crowding distance to each solution. The crowding distance value is based on the position in the front and is used to ensure that solutions do not coagulate around a particular solution subspace.
The sorting algorithm is defined below by Liong et al. (2004) and summarized by Chang &
van Zyl (2012):
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If one considers 2 unique solutions (chromosomes) within the initialized population:
x(1) & x(2),
i. x(1) is better than x(2) if x(1) is no worse than x(2) in all objectives & at least one objective is better;
ii. x(2) is better than x(1) if x(2) is no worse than x(1) in all objectives & at least one objective is better;
iii. Otherwise, x(1) & x(2) are equally good.
This process is repeated until all chromosomes are ranked by domination. Crowding distance will be demonstrated later in the methodology.
The population is entered into the evolution process. The parents are selected based on the fitness probability and the reproduction process starts. The crossover process, which in NSGA-II is the Simulated Binary Crossover (SBX), ensures that two solutions with fit decision variables (genes) are combined to form a solution with a high probability to have a better fitness than the parents. To maintain diversity within the population, a certain probability exists with the possibility for a mutation to occur (±10%) of a decision variable that has a low probability for selection, i.e. an unfit chromosome.
Selection is performed on the population which now contains both the parents and the child solutions and the unfit individuals are replaced by the fit ones to maintain a constant population size (Seshadri, 2009). This process is iterated with the new chromosome population being entered back into the evolution process until the specified number of generations has been reached.
The output from the genetic algorithm (GA) is a Pareto-optimal front that represents a trade off curve between the objectives set at the initial stages. The real advantage of multi- objective GAs is seen at this point is that the front provides flexibility to the engineer upon which he can apply his/her engineering insight to provide a practicable solution to the client (Prasad & Park, 2004). This is discussed in more detail in the methodology.
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