An Exact Algorithm for Bilevel 0-1 Knapsack Problems

Algorithm 2

Forward procedure for the leader.

𝑓¹₁(𝜗)=0 if 𝜗=0, 𝑑¹₁ if 𝜗=𝑎¹₁, −∞ otherwise;

 for  𝑘=2 to 𝑛¹  do

     for  𝜗=0 to 𝑏  do

    if 𝜗<𝑎¹_𝑘  then

      𝑓¹_𝑘(𝜗)=𝑓¹_𝑘−1(𝜗);

    end

    else

     𝑓¹_𝑘(𝜗)=max(𝑓¹_𝑘−1(𝜗),𝑓¹_𝑘−1(𝜗−𝑎¹_𝑘)+𝑑¹_𝑘);

    end

     end

 end