Select matching raws in table with Minizinc

Question

I have a table of Projects with 4 features (name, value, cost, people). I want to describe a model in Minizinc that selects the projects that maximize the total value but with a total budget of 255 and I can choose almost 9 projects between 20.

I write a data file: data.dzn

%       name       value    cost  people
data = [
      | PRJ0,      6000,    35,    5
      | PRJ1,      4000,    34,    3
      | PRJ2,      1000,    26,    4
      ...
      | PRJ20,     1200,    18,    2    
      |];


PRJ = {PRJ0,...,PRJ20};
FEATURE = {name, value, budget, personnel};
max_budget = 225;
max_prj=9;

So my constraints are:

choose_project <= 9 /\ budget<=255 s.t solve maximize tot_value;

How can I select a non-arbitrary number (1 to 9) of decision variables among projects raw in the table? Until now this is my code: invest.mzn

include "data.dzn";
int: max_budget;  %255
int: max_prj;     %9
enum FEATURE;
enum PRJ;
array[PRJ,FEATURE] of int: data;
constraint ...
...
solve maximize tot_value;

Answer 1

You can declare an array of Boolean variables, say selected_projects, that encodes whether or not a project PRJ_k is selected or not.

Then you can simply count how many projects in this array are being selected at the same time.

Example:

enum FEATURE = {name, value, budget, personnel};
enum PRJ = { PRJ0, PRJ1 };

array[PRJ,FEATURE] of int: data =
     [| PRJ0,      6000,    35,    5
      | PRJ1,      4000,    34,    3
      |];

array[PRJ] of var bool: selected_projects;

var int: tot_value;

% The total number of selected projects must be in [1, 9]
constraint let {
        var int: tot_selected = sum(prj in PRJ) ( selected_projects[prj] )
    } in
        1 <= tot_selected /\ tot_selected <= 9;

constraint tot_value = sum(prj in PRJ) ( selected_projects[prj] * data[prj, value] );

% ...
% encoding of budget and personnel constraints
% ...

solve maximize tot_value;