Candidate solution

DENDRAL followed a three-stage procedure. In the first phase, the so-called plan, prior knowledge, and heuristics were used to deduce a set of constraints. Constraints could be, for example, the exemption of large sets of candidate solutions or the suggestion for a extensive search over limited classes of solutions. 

Representation requires that the designer of a typical evolutionary computation algorithm (EA) formulates one inadaptable blueprint for the solution of some problem, then present the variables of that blueprint in a form that is amenable to manipulation by the genetic operators of the EA. Fitness evaluation, on the other hand, has limited GA in two distinct ways (1) it has limited environmental feedback to the confines of a formula or algorithm, which reflects accurately and exclusively the quality of the complete candidate solution from the perspective of the human designer. In addition, (2) fitness evaluation has proven to be the most computationally costly part of a typical EA. Note that elaborate developmental mappings actually increase that computational cost. However, our interest here lies in the limiting effects of representation. 

A standard branch-and-bound algorithm is used to explore the first-stage search space while lower bounds are provided by the Lagrangian dual. Candidate solutions... 

The hybrid evolutionary algorithm for 2S-MILPs is realized by using an evolution strategy (ES) to solve the master problem of the intensive 2S-MILP. Each individual of the ES represents a first-stage candidate solution x. The object parameters are encoded by a mixed-integer vector. The fitness of an individual is evaluated by the objective function of the master problem (MASTER),/ (x). 

The last entry in Table 1.1 involves checking the candidate solution to determine that it is indeed optimal. In some problems you can check that the sufficient conditions for an optimum are satisfied. More often, an optimal solution may exist, yet you cannot demonstrate that the sufficient conditions are satisfied. All you can do is show by repetitive numerical calculations that the value of the objective function is superior to all known alternatives. A second consideration is the sensitivity of the optimum to changes in parameters in the problem statement. A sensitivity analysis for the objective function value is important and is illustrated as part of the next example. 

Since an exhaustive search—eventually combined with exhaustive evaluation— is practically impossible, any variable selection procedure will mostly yield subopti-mal variable subsets, with the hope that they approximate the global optimum in the best possible way. A strategy could be to apply different algorithms for variable selection and save the best candidate solutions (typically 5-20 variable subsets). With this low number of potentially interesting models, it is possible to perform a detailed evaluation (like repeated double CV) in order to find one or several variables... 

Select any candidate solution x1N,..., xJJ obtained from the previous steps. However, it is preferable to select the optimal solution with the minimum objective function value, that is ... 

Note that due to sampling error we may find that Vn< < vn- For this reason the confidence interval obtained by (7.22) provides a more conservative bounding. The above procedure for the validation of a candidate solution was originally suggested by Norkin, Pflug and Ruszczysk (1998) and further developed by Mark, Morton and Wood (1999). 

The set of candidate solutions considered by a search procedure is often called the search space of the problem. For molecular design problems, there are several possible search spaces, the most common being sequence space, the space of all molecules being considered [4,37-39], The concept of a sequence space is important because it provides a framework for formal theory and it has heuristic value in developing intuition for searches and communicating ideas. Sequence spaces are discrete, though search spaces in general may be discrete, continuous, or discrete on some axes and continuous on others. 

An initial candidate solution (the first parent) is created either entirely at random or, as in the case of the GA, by using available domain-specific information about the solution space. 

An EA works with a set of candidate solutions to the optimization problem. A solution is referred to as an individual and a set of p solutions is called the population. Each individual has a fitness value which shows how good the solution is with respect to the objective function. X new individuals are added to the population by recombination and mutation of existing individuals. The idea is that the new individuals inherit good characteristics from the existing individuals. The X worst solutions are removed from the population. After several iterations, which are called generations, the algorithm provides a population that comprises good solutions. 

In our case, the EA generates individuals which represent a fully ordered sequence of the recipe steps of all batches. To evaluate the fitness of the candidate solutions, the assignment of the equipment, the routing of the AGVs and a calculation of the durations of the recipe steps are carried out by the simulation algorithm of [1]. The recipe steps are identified by their batch IDs to maintain precedence relations in the recipes when... 

Empirical observations with exhaustive searches sug sted 2o to be a reasonable cutoff for the assessment of the goodness of any candidate solution. 

Evolutionary algorithms (EAs) have been successfully applied to a range of multi-objective problems. They are particularly suitable for multiobjective problems as they result in a set of non-dominated solutions in a single run. Furthermore, EAs do not rely on functional and slope continuity and thus can be readily applied to optimization problems with mixed variables. However, EAs are essentially population based methods and require evaluation of numerous candidate solutions before converging to the desired set of solutions. Such an approach turns out to be computationally prohibitive for realistic Multidisciplinary Design Optimization problems and... 

An external archive of actual evaluations is maintained and used to create the surrogate models. The archive only maintains the unique candidate solutions evaluated using actual evaluations over generations. 

A surrogate model is created for each of the objectives and the constraints using a fraction (0 < a < 1) of the candidate solutions in the... 

The procedure for selection of parents is the same as that of NSGA-II. Binary tournament is used to select a parent from two individuals. Binary tournament between two candidate solutions x and x is performed as follows ... 

In order to maximize the use of information from all actual evaluations, the algorithm maintains an external archive that is used to train the RBF model, periodically after every S generations. In order to maintain prediction accuracy, a candidate solution is only approximated using the... 

RBF model if at least one solution exists in the archive that is within a distance threshold. This distance threshold plays an important role in the early stages of evolution where more candidate solutions are evaluated using actual computations even during the surrogate phase. Furthermore, a candidate solution is only evaluated using the surrogate model if the MSE of the surrogate on the validation set is below an user defined threshold. 

The results of all the test problems support the fact that better non-dominated solutions can be delivered by the SAEA as compared to NSG A-II for the same number of actual function evaluations. Although the algorithm incurs additional computational cost for solution clustering and periodic training of RBF models, such cost is insignificant for problems where the evaluation of a single candidate solution requires expensive analyses like finite element methods or computational fluid d3mamics. 

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