A.50.4.2 Example 2

Here is an instance of the knapsack problem described above, where C = 8, and we have two types of items: One item with value 7 and size 6, and 2 items each having size 4 and value 4. We introduce two variables, x(1) and x(2) that denote how many items to take of each type.

:- use_module(library(simplex)).

knapsack(S) :-
        knapsack_constraints(S0),
        maximize([7*x(1), 4*x(2)], S0, S).

knapsack_constraints(S) :-
        gen_state(S0),
        constraint([6*x(1), 4*x(2)] =< 8, S0, S1),
        constraint([x(1)] =< 1, S1, S2),
        constraint([x(2)] =< 2, S2, S).

An example query yields:

?- knapsack(S), variable_value(S, x(1), X1),
                variable_value(S, x(2), X2).
X1 = 1
X2 = 1 rdiv 2.

That is, we are to take the one item of the first type, and half of one of the items of the other type to maximize the total value of items in the knapsack.

If items can not be split, integrality constraints have to be imposed:

knapsack_integral(S) :-
        knapsack_constraints(S0),
        constraint(integral(x(1)), S0, S1),
        constraint(integral(x(2)), S1, S2),
        maximize([7*x(1), 4*x(2)], S2, S).

Now the result is different:

?- knapsack_integral(S), variable_value(S, x(1), X1),
                         variable_value(S, x(2), X2).

X1 = 0
X2 = 2

That is, we are to take only the two items of the second type. Notice in particular that always choosing the remaining item with best performance (ratio of value to size) that still fits in the knapsack does not necessarily yield an optimal solution in the presence of integrality constraints.