Objective Function Evaluation

Randomly initialize the population and evaluate objective functions and constraints of each individual in the population by calling HEN model . ... 

The great advantage of the evaluated objective function is the fact that no constraints are necessary which decreases the computational effort during optimization considerably. 

Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues. First, one must have a representation of the artifact that can be used to determine how the artifac t performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself. Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions. Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this sec tion. 

Objective Provide a basis to evaluate the functional capabilities, design limitations, and operational requirements of the system, and to evaluate the vendor s ability and willingness to support the system. 

The golden section search is the optimization analog of a binary search. It is used for functions of a single variable, F a). It is faster than a random search, but the difference in computing time will be trivial unless the objective function is extremely hard to evaluate. 

Here (j is the CG update parameter. In the above equations, e = e (tj) o vector notation for the discretized electric field strength, = g (fj) o objective functional J with respect to the field strength (evaluated at a field strength of e t) and dk = d (t ) o search direction at the feth iteration. The time has been discretized into N time steps, such as that tj=jx )t, where j = 0,1,2, , N. Different CG methods correspond to different choices for the scalar (j. ... 

Step 4 Evaluate the objective functional according to Eq. (1). Note that the last term in the equation is zero, as is a solution of the time-... 

In this appendix we follow the treatment of Maday and Turinici [94], and show that the iterative scheme laid out in Eqs. (21.a-d) is guaranteed to converge. The convergence of the algorithm can be proved by evaluating the difference between the values of the objective functional between two successive iterations. Suppose that 5 0 and t 0, then,... 

Uncertainties in amounts of products to be manufactured Qi, processing times %, and size factors Sij will influence the production time tp, whose uncertainty reflects the individual uncertainties that can be presented as probability distributions. The distributions for shortterm uncertainties (processing times and size factors) can be evaluated based on knowledge of probability distributions for the uncertain parameters, i.e. kinetic parameters and other variables used for the design of equipment units. The probability of not being able to meet the total demand is the probability that the production time is larger than the available production time H. Hence, the objective function used for deterministic design takes the form ... 

Direct search methods use only function evaluations. They search for the minimum of an objective function without calculating derivatives analytically or numerically. Direct methods are based upon heuristic rules which make no a priori assumptions about the objective function. They tend to have much poorer convergence rates than gradient methods when applied to smooth functions. Several authors claim that direct search methods are not as efficient and robust as the indirect or gradient search methods (Bard, 1974 Edgar and Himmelblau, 1988 Scales, 1986). However, in many instances direct search methods have proved to be robust and reliable particularly for systems that exhibit local minima or have complex nonlinear constraints (Wang and Luus, 1978). 

Minimization of S(k) can be accomplished by using almost any technique available from optimization theory, however since each objective function evaluation requires the integration of the state equations, the use of quadratically convergent algorithms is highly recommended. The Gauss-Newton method is the most appropriate one for ODE models (Bard, 1970) and it presented in detail below. 

Once an acceptable value for the step-size has been determined, we can continue and with only one additional evaluation of the objective function, we can obtain the optimal step-size that should be used along the direction suggested by the Gauss-Newton method. 

If we wish to avoid the additional objective function evaluation at p=qa/2, we can use the extra information that is available at p=0. This approach is preferable for differential equation models where evaluation of the objective function requires the integration of the state equations. It is presented later in Section 8.7 where we discuss the implementation of Gauss-Newton method for ODE models. 

The main difference with differential equation systems is that every evaluation of the objective function requires the integration of the state equations, In this section we present an optimal step size policy proposed by Kalogerakis and Luus (1983b) which uses information only at g=0 (i.e., at k ) and at p=pa (i.e., at... 

Having evaluated the system performance for each setting of the six variables, the variables are optimized simultaneously in a multidimensional optimization, using for example SQP, to maximize or minimize an objective function evaluated at each setting of the variables. However, in practice, many models tend to be nonlinear and hence a stochastic method can be more effective. 

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). 

Process synthesis is a task of formulating the process configuration for a purpose by defining which operations or equipment are used and how they are connected together. There are two basic approaches for process synthesis 1) classical process synthesis, analysis and evaluation, and 2) optimization of process structure by using a suitable objective function. 

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