An Exact Algorithm for Bilevel 0-1 Knapsack Problems

Algorithm 1

A polynomial time solution procedure for CBKP.

 Let 𝑑_opt denote the optimal value of the leader objective function;

 Let (𝑥_opt,𝑦_opt) be an optimal solution of CBKP;

 Initialization

  Sort the variables 𝑥 and 𝑦 in decreasing order of the ratio 𝑑¹_𝑗/𝑎¹_𝑗 for 𝑥 and 𝑐_𝑗/𝑎²_𝑗

  and 𝑑²_𝑗/𝑎²_𝑗 for 𝑦;

  Let

     𝑥^∗₁=1,𝑥^∗₂=1,…,𝑥^∗_𝑘=1,𝑥^∗_𝑘+1=(𝑏−∑^𝑘_𝑖=1𝑎¹_𝑖)𝑎¹_𝑘+1,𝑥^∗_𝑘+2=0,…,𝑥^∗_𝑛1=0

  be the optimal solution for the following problem:

          𝑥^∗∈Argmax{𝑓¹(𝑥)=𝑑¹𝑥∶𝑎¹𝑥≤𝑏,0≤𝑥≤1};

  Let

    𝑦^∗₁=1,𝑦^∗₂=1,…,𝑦^∗_𝑡=1,𝑦^∗_𝑡+1=(𝑏−𝑎¹𝑥^∗−∑^𝑡_𝑖=1𝑎²_𝑖)𝑎²_𝑡+1,𝑦^∗_𝑡+2=0,…,𝑦^∗_𝑛2=0

  be the optimal solution for the following problem:

         𝑦^∗∈Argmax{𝑓²(𝑦)=𝑐𝑦∶𝑎²𝑦≤𝑏−𝑎¹𝑥^∗,0≤𝑦≤1};

  /*   The indexes 𝑘+1 and 𝑡+1  are respectively the indexes of the last

  leader and follower variables with a positive value according

  to the ordering */

  Let 𝑑_opt=𝑑¹𝑥^∗+𝑑²𝑦^∗;

 𝑖=𝑘+1; 𝑗=𝑡+1;

 while  (𝑖>0) and (𝑗<𝑛²+1) do

  if (𝑎¹_𝑖𝑥^∗_𝑖>𝑎²_𝑗(1−𝑦^∗_𝑗))  then

   𝑥^∗_𝑖=𝑥^∗_𝑖−𝑎²_𝑗(1−𝑦^∗_𝑗)/𝑎¹_𝑖; 𝑦^∗_𝑗=1; 𝑗=𝑗+1;

  end

  else  if (𝑎¹_𝑖𝑥^∗_𝑖<𝑎²_𝑗(1−𝑦^∗_𝑗))  then

    𝑥^∗_𝑖=0; 𝑦^∗_𝑗=𝑦^∗_𝑗−𝑎¹_𝑖𝑥^∗_𝑖/𝑎²_𝑗; 𝑖=𝑖−1;

  end

  else  if (𝑎¹_𝑖𝑥^∗_𝑖=𝑎²_𝑗(1−𝑦^∗_𝑗))  then

    𝑥^∗_𝑖=0; 𝑦^∗_𝑗=1; 𝑖=𝑖−1; 𝑗=𝑗+1;

  end

  if (𝑑_opt<𝑑¹𝑥^∗+𝑑²𝑦^∗)  then

   𝑑_opt=𝑑¹𝑥^∗+𝑑²𝑦^∗; (𝑥_opt,𝑦_opt)=(𝑥^∗,𝑦^∗);

  end

 end