linear programming - Algorithm of choosing several items with best characteristics to maximize profit -

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let's have items a1-a10 different characteristics. example a1 (x1=4; x2=2; x3=1; x4=3). , have 5 categories c1-c5 put items into. 

x1 - matters category 1,  x2 - category 2,  x3 - category 3,  x4 - categories 4 , 5.

it must @ least 1 item in each category.

our goal put items different categories way - output maximum.

example:

let's have 4 items a1 (x1=3; x2=1; x3=1), a2 (x1=2; x2=1; x3=2);  a3 (x1=1; x2=1; x3=3); a4(x1=1; x2=3; x3=1) , 3 categories c1(x1);  c2(x2); c3(x3).

here best way put a1 category c1 (+3 on output); a3 category c3 (+3 on output); a4 category c2 (+3 on output) , a2 either category c1 or c3 (+2 on output). our output 11.

best solution of given example.

here's picture bad solution of given example output 4. worst solution of given example.

but easy example. have complex solve.


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