$title Gear Train Design (GEAR,SEQ=203)

$onText
A compound gear train is to be designed to achieve a specific
gear ratio between the driver and driven shafts. The objective
of the gear train design is to find the number of teeth of the
four gears and to obtain a required gear ratio of 1/6.931.

Deb, K, and Goyal, M, Optimizing Engineering Designs Using a
Combined Genetic Search. In Back, T, Ed, Proceedings of the
Seventh International Conference on Genetic Algorithms. 1997,
pp. 521-528.

Keywords: mixed integer nonlinear programming, engineering, gear train designment
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option optCr = 1e-5;

Integer Variable
   a 'number of teeth in each of the gears (integer)'
   f 'number of teeth in each of the gears (integer)'
   b 'number of teeth in each of the gears (integer)'
   d 'number of teeth in each of the gears (integer)';

Variable z 'objective variable';

Equation
   obj  'objective function'
   sym1 'symmetry'
   sym2 'symmetry';

Scalar
   invgratio 'reciprocal of required gear ratio' / 6.931 /
   minteeth  'minimum number of teeth'           / 12    /
   maxteeth  'maximum number of teeth'           / 60    /;

a.lo = minteeth; a.up = maxteeth;
f.lo = minteeth; f.up = maxteeth;
b.lo = minteeth; b.up = maxteeth;
d.lo = minteeth; d.up = maxteeth;

obj..  z =e= 1 + sqr(invgratio - (a*f)/(b*d));

sym1.. d =g= b;

sym2.. a =g= f;

* A better starting point than just the lower bounds
a.l = (maxteeth - minteeth)/2;
f.l = (maxteeth - minteeth)/2;
b.l = (maxteeth - minteeth)/2;
d.l = (maxteeth - minteeth)/2;

Model m1 / all /;

solve m1 using minlp minimizing z;

Scalar
   calcinvratio 'ratio of solution'
   error        'absolute error';

calcinvratio = (a.l*f.l)/(b.l*d.l);
error        = abs(calcinvratio - invgratio);

display calcinvratio, invgratio, error;