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roller bearings must be greater than 15 mm.
Constraint group 23 The shearing and crushing stresses
must not exceed a specif ied value on the keys and keyways
used the attach the four gears to the shafts.
Constraint group 24 The operating temperature of the
reducer must not exceed a specif ied value.
5 A mass minimization problem
Let us now consider the following design problem. A 2.9 kW two-stage reducer is to be designed for minimum weight and a service life of 8,000 h, given an input speed of 925 RPM and a transmission ratio of 7.6. The gears should be based on an ISO 53 basic rack profile, with the pinions and wheels made of quenched and tempered 42CrMo4 and 41Cr4 respectively.
Running the algorithm described earlier yielded a reducer with a 2.8 × 2.7 pision of the transmission ratio and axial distances of 80 and 100 mm on stages one and two respectively, weighing 44.3 kg. This solution was found on the boundary of the second transmission stage Hertzian contact pressure constraint, very near to four additional constraint boundaries. These are highlighted in Fig. 2.
In contrast to the successful determination of this global optimum found in an ‘awkward’ corner defined by several constraint boundaries, consider the outcomes of a set of benchmark experiments. The natural basis for comparison is, of course, the standard GA the multi-epoch heuristic is built upon, run with the epoch switching feature disabled. Multiple repeated runs of this GA failed to reach the objective value of 44.3 kg within a budget of 300,000 evaluations. In fact, this standard GA failed to reach even a slightly relaxed threshold objective value of 45.8 kg by an arbitrarily
selected cut-off point of 300,000 evaluations. By comparison, the multi-epoch algorithm has attained this value on
Fig. 2 The plus symbols indicate the values of the constraints of the problem at the optimum design, with black dots highlighting the five constraints whose boundaries are closest to that optimum (note that the value gi of constraint number i is defined as gi = ai /bi −1, where the constraint is of the form ai < bi ) every one of 50 independent runs (started from different random initial populations) of up to 300,000 evaluations— Fig. 3 is a histogram of the number of evaluations required by it to do so (average of just under 75,000 evaluations, standard deviation of just over 54,000). Perhaps even more tellingly, as the first bar of the histogram shows, 17 of the 50 runs passed the threshold weight value after less than 50,000 evaluations.
As an additional benchmark, we have also tested a Simulated Annealing optimizer (an implementation of a recent version of the heuristic by Talbi 2009) as a means of tackling the reducer problem. Experiments with five different types of cooling schedules (linear, exponential, parabolic, hyperbolic and power) run to 1,000,000 evaluations of the objective function, failed, just as in the case
of the “plain” Genetic Algorithm, to reach the threshold weight, once again underscoring the extreme difficulty of this mixed variable problem.
In addition to the solution of the basic design problem, the type of optimization capability demonstrated here opens the possibility of evaluating objective function sensitivities with respect to the elements of the design brief in a timely manner. Consider, for example, the impact of the required service life of the reducer on the mass (our objective function in this study). This is a high level relationship shrouded by a plethora of low level connections between the design variables and the constraints, whose analytical calculation can be considered, for all practical purposes, intractable.
We can, however, obtain discrete handholds on this relationship by simply running the optimizer for different values of the service life—the results of such a study are shown in Fig. 4.