Non-linear Programming Problems with Mathematical Formulations
Various non-linear programming problems involving maximizing triangle area, square and circle combination, and cylinder volume, alongside a water supply project optimization scenario. Mathematical formulations and solutions are provided for each problem.
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Presentation Transcript
Triangle with Maximum Area Assume that there is a metal wire with length L and we asked to make a right angle triangle with this wire with maximum area. Formulate the problem as an NLP. Max St ab + 12 a a a c + = b + c L a = 2 2 2 b b c L , , 0 c b
Square and Circle Assume that there is a metal wire with length L and you asked to make a circle and a square with maximum total area. Formulate the problem as an NLP. + + x 2 2 Max St x r r = 4 2 x r r L L , 0 x
Cylinder With Maximum Volume Assume that we want to make a cylinder tank by an a b metal sheet with maximum volume. Formulate the problem as a non-linear programming problem. 2 r 2 Max St r h r r + 2 h 2 r 2 rh ab h a , 0 r b r h
Water Supply Project In a water supply project the water will pump from a dame to the five villages. To do this we will use a big water pump. Now the aim is to find the coordinate of the pump where its distance from the dame be minimize and its distance from each of the village be not more that 40 miles. The following figure shows is the coordinate of the dame and the villages. ( , ) a b ( , x y ) 5 5 + ( ) ( ) 2 2 Min x a y b ( , ) x y + . ( ) ( ) 40 2 2 S t x x y y ( , x y ) 1 1 ( , x y ) 1 1 4 4 + ( ) ( ) 40 2 2 x x y y 2 2 + ( ) ( ) 40 2 2 x x y y 3 3 ( , x y ) + ( ) ( ) 40 2 2 x x y y 3 3 4 4 ( , x y ) + 2 2 ( ) are free ( ) 40 2 2 x x y y 5 5 , x y
2 + + + 200 150 200 300 Min D D D D 1 2 3 4 + = . ( 5) ( 10) 2 2 S t x y D 1 + = ( 10) ( 5) 2 2 x y D 2 + = ( 0) ( 12) 2 2 x y D 3 + = ( 12) ( 2) 0 2 2 x y D 4 = , 1,2,3,4 x y are free D for i i
