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The sensitivity analysis conducted for minimization of the function of reliability index has led to the following appropriate values to be applied for GA analysis: probability of crossover, Pc = 0.65, initial population, Npop = 50, mutation probability, Pm = 0.025, and maximum number of generation, MaxGen = 40.
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In these cases, the estimation of the model parameters comprises numerical iterative methods for minimization of the objective function.
A scalar parameter was proposed as a tuning knob for minimization of the sensitivity function and uncertainty conditions.
An iterative consensus-like algorithm is applied for minimization of the cost function, similar to the one described in (Ghaderi and Srikant, 2013).
Two different types of co-ordinate functions are used: sinusoids, and polynomial expressions which contain an undetermined exponential parameter which allows for minimization of the natural frequency coefficients.
Open image in new window Fig. 1 Judgment of trial vector for minimization of single variable function.
(1), (2), and (3) are for the objective function for minimization of total distance traveled by all vehicles, the objective for minimization of total number of vehicles used, and the vehicle capacity constraint, respectively.
For the minimization of the objective function a particle swarm algorithm was employed.
Using an adjoint equation we implement a gradient algorithm for the minimization of the objective function.
The observed output error is used for the minimization of the cost function by calculating the value of Eq. (4).
Table 2 provides a summary of the optimal parameters as calculated from the Singular Value Decomposition for the minimization of the cost function of Eq. (3).
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